RECENT DEVEL oPMEnT ON COMMUTATIVITY OF RINGS & ITS APPLICATIONS

1. 4th COLLOQUIUM LECTURE RECENT DEVELoPMEnT ON COMMUTATIVITY OF RINGS & ITS APPLICATIONS By Moharram Ali Khan,PhD (8th September, 2016) • Professor of Mathematics • Specialization: • Commutativity theorems for rings/near-rings( 2010 MSC; 16U80: 16Y70) • Derivations on rings/near-rings/Banach algebras( MSC;16N10: 16W25) • Mathematical modeling (2010 MSC: 60K25) RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

2. CONTENTS • ABSTRACT • ORIGIN OF RING THEORY • GENERALIZATIONS & EXTENSIONS • COMMUTATIVITY THEOREMS:4.1 Commutativity theorems for rings 4.2 Generalizations of commutativity theorems for rings 5. SPECIAL CLASS OF RINGS: 5.1 s-unital ring; 5.2 Periodic ring; 5.3 Ring properties ; 5.4. Open problems (Questions) 6. ORIGIN OF NEAR RINGS 7. COMMUTATIVITY THEOREMS FOR NEAR-RINGS 8. EXTENSIONS OF COMMUTATIVITY OF NEAR- RINGS 8.1 Near-ring Properties: 8.2 Problems 8.3 Conclusions 9. APPLICATIONS OF NEAR-RING: 9.1 Construction of planar near-rings 10. PROJECT: 10.1 Ring properties; 10.2 Questions 10.3 Near-rings: 10.3.1 Algebraic properties ; 10.3.2 Differential properties ; 10.3.3 Questions RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

3. 1. ABSTRACT In this lecture, we discuss the recent contributions on commutativity theorems for rings and its application in near-rings. In addition, we justify our results by motivations together with examples. Finally, we close our discussion with some open problems for future endeavors. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

4. 2. ORIGIN OF RING THEORY • The origin of ring theory lies in the work of Dedekind. • In 1870, Dedekind proved that • the factorization properties of integers extend to algebraic integers • if elements are replaced by ideals and gave the words “ring”, “ideal”, • and “field”. • The Modern Theory of rings was developed in 1920s by • Amalie Emmy Noether (1882-1935). • In 1921, Noether [in her famous paper “The theory of ideals and • rings”] established the following. • A ring R is a “system” closed under two abstract operations ‘+’ and • ‘’, to which she gives the names addition and multiplication; • these operations are required to satisfy six axioms: RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

5. The associative law of addition • (x + y) + z = x+ (y + z). • The commutative law of addition • x+ y = y + x. • The associative law of multiplication • (x  y)  z = x (yz). • The commutative law of multiplication • xy = yx • The distributive law, for multiplication over addition • x (y + z) = x  y + x  z. • For any a and b in R, there exists a uniqueelement x satisfying the equation • a + x = b. • From this definition, the study of rings was transformed into a powerful • abstract theory, one of the pillars of modern mathematics. • Todaycommutativity of multiplication is not part of the definition of a ring. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

6. Emmy Noether also made the notion of “ideal” a central concept in her exposition, framing it in a general setting: • “Ideal of a ring R is a nonempty subset I such that if x and y belongs in I, then x-y, • r x and x rI for any r in R”. • It is worth pointing out that Noether’s definition of a ring is not the one in common use today; the current one usually specifies that • R is a commutative group under addition. • But this is ensured by her VI axiom, as demonstrated in the following: • Take an arbitrary, but fixed, element a in R. Then the equation a + x = a has a • solution in R; denote it by 0. • For any other element b in R, if z is the solution of the equation a + x = b , then • z + a = a + z = b, and b + 0 = (z + a) + 0 = z + (a + 0) = z + a = b, making • 0 an identity element for the operation of addition. • Solution of a + x = 0 will furnish the additive inverse of a, and it is denoted by ‘-a’. • Historically, many of the fundamental notions in Emmy Noether’s abstract theory of • ideals can be traced to the work of Dedekind. • In 1930, Vander Warden's Modern Algebra, abstract algebra become one of the • main branches of mathematics. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

7. 3. Generalizations and Extensions “…A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas.” [G.H. HARDY (1877- 1947)] RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

8. 4. COMMUTATIVITY THEOREMS Since the late of 1940’s shortly after the development of the general structure theory for associative ring (i.e. its application say to Commutativity theorems [see for reference; Jacobson “Structure of Rings” AMS, 1964]. Recently there are several results dealing with conditions under which ring is commutative. Generally, such conditions are placed on the ring itself or its subsets, or its commutators. • The aim of this lecture is: To investigate a necessary condition under which an arbitrary ring (ring with unity, s-unital or periodic ring) becomes commutative. • This sort of problem has attracted the attention of many mathematicians. Among them are Jacobson, Kaplansky, Herstein, Tominaga, Bell, Hirano etc., who have contributed immensely to develop the theory of rings. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

9. 4.1 Commutativity theorems for rings Commutativity theorems are part of the study of polynomial identities in non-commutative rings. They are theorems which assert that, under certain conditions, the ring at hand must be commutative. The proofs of theorems of this sort in their general form require the structure theory for non-commutative rings. Instances of these theorems have a strongly computational flavor. They provide interesting test examples for algorithms which use rewrite rules and reduction theory for polynomial rings in non-commuting variables. • In Small’s collection of review of ring theory; • commutativity theorems occupy five of the six subsections of section 28. • In 1940 – 1979;this occupies pages 827-842 and comprises 117 papers. • In 1980- 1984; MR classification 16U80 ; titled commutativity theorems, • occupies pages 455-468 and comprises 102 papers. • James Pinter-Lucke (2007);Commutativity conditions for rings: 1950–2005 RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

10. 4.2 Generalizations of commutativity theorems for rings In 1905 [TAMS],Wedderburn: A finite division ring is necessarily a field. • This result has attracted the imagination of most mathematicians because it is • so unexpected, interrelating two seemingly unrelated things, namely the • number of elements in a certain algebraic system and the multiplication of the • system. • For algebraists the Wedderburn theorem has served as a jumping-off point for • a large area of research, in the 1940s and 1950s concerned with the • commutativity of rings. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

11. In 1946 [Ann. of Math],Jacobson (J-ring): If R is a ring in which for every xR, there exists an integer n(x) > 1 such that xn(x) = x, then R is commutative Jacobson pointed out that “we were interested to prove it for fixed value of n, but during the conversation with Prof. Kaplansky, who suggested establishing for general value of n”. In 1951[Amer.J.Math], Kaplansky (K-ring): A ring R is called a k-ring if and only if for every xR, there exists an integer n(x) > 1 such that xn(x)Z(R),center of R, then R is commutative. In 1953 [Amer. J. Math], Herstein [H-ring]: Let R be a ring satisfying the property for every xR, there exists an integer n(x) > 1 such that xn(x) -xZ(R). Then R is commutative. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

12. In 1986 [Bull. Austral. Math.Soc.], Khan et al.: We proved that if R is a semi prime ring satisfying the condition [(xy)2 – xy, x] = 0, for all x, y in R, then R is commutative. The ring of 3×3 strictly upper (or lower) triangular matrices over an associative ring satisfies the hypothesis of theorem but need not be commutative. This indicates that the theorem is not valid for arbitrary rings. In 1975 [Canad. J. Math], Bell: A ring R is commutative if for each pair of elements x, y R, there exist integers m, n  1 such that xy = ymxn. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

13. In 1988 [Math. Japon.] Khan et al.: An associative ring R is commutative if a ring with identity 1 satisfying the property, for each pair of elements x, y R, there exist positive integer m > 1, n  1 such that [xy - ymxn, x] = 0. We gave an example to justify the hypothesis of this result are not superfluous: Take Dk = { (aij)k×k matrices over D} and Ak = {(aij) k×kDk.aij =0, i j}. Ak is non commutative ring for any k > 2. For instance, A3 satisfies [xy -ymxn, x] = 0, for all m and n. In this paper, we posed a question: It would be interesting to prove this result for m and n depend on x and y. In 1991 [Math. Japon.] Kezlan]: If n is a positive integer and a ring R with unity1 satisfying the property: Given for each pair of elements x, y R there exists an integer m = m(x, y) > 1 such that [xy - ymxn, x] = 0, then R is commutative. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

14. In 1993 [Studia Sci. Math.Hungar.] Bell et al: A semi prime ring R must be commutative if for each x  R,  an integer n = n(x) > 1 such that (xy)n– xn yn is central for each y in R. In 1995 [Studia Sci. Math.Hungar.] Khan jointly with Bell et al: Let R be a ring with no non zero nil ideals. If for each x, y  R,  an integer n = n(x, y)  1 such that [(xy)n– xn yn , x] = 0 =[(yx)n– xn yn , x], then R is commutative. It would seem that the human race now has enough Commutativity theorems, at least for a while. Now, I am about to present the following class of rings. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

15. 5. SPECIAL CLASS OF RINGS In an effort to have a class of rings that are close to having unit, the concept of s-unital was developed. 5.1 s-unital rings A ring R is called left (resp. right) s-unital if x  Rx (resp. x xR) for each x  R. Further, R is called s-unital if it is both left as well as right s-unital. If R is s-unital (resp. left, or right s-unital), then for any finite subset F if there exists an element e in R such that ex = x e = x (resp. ex = x or x e = x), for all x F. Such an element e is called a pseudo (resp. Pseudo left or pseudo right) identity of F in R. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

16. 5.2 Periodic Ring • A ring R is called periodic if for each xR the set {x1, x2,x3,….} is finite. • In other words; for each xR there exists distinct natural numbers m(x), n(x) • such that x m(x) = x n(x). • The concept of Boolean ring was first introduce by Stone [TAMS; 1936], as a • ring in which every element is idempotent. • In 1937, McCoy & Montgomery [Duke Math. J.] introduced the concept of p-ring , p a prime, as a ring R in which xp= x and • p x =0, for all xR. • Boolean rings are simply 2-rings ( p = 2). • A concept that generalizes Boolean rings and p-rings is that of a periodic ring. • Examples of periodic rings are nil rings. • For thirty years various author studied commutativity in rings satisfying the polynomial identity of the form • [(xy)m = xm ym, m >1 [see for reference Bell; Math. Mag. (1982)]. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

17. 5.3 Ring properties Let m  1 be a fixed positive integer and S a non-empty subset of R. Now we consider the following ring properties: For any x, y in S, C1 ( m, S) [xm, ym] = 0 C2( m, S) [x, [x, ym] ] = 0 C3 ( m, S) (xy)m = xmym C4( m, S) (xy)m xmym  Z(R) C5( m, S) (xy)m ymxm  Z(R) C6 m, S) (xy)m ymxm, x]   =[(yx)m  xmym, x] C7 m, S) (yx)m xm xm(xym, x]   Q(m) For any x, y in R, m [x, y] = 0  [x, y] = 0. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

18. In 2003 [Czechoslovak Math.J] Khan : Let R be a ringwith 1satisfiesC1 (m, R/J(R)),C6 m, R/J(R)) and Q(m). Then R is commutative.CzechMathJ_53-2003-3_5.pdf In continuation of this paper, one can ask a natural question: What can we say about commutativity of R if the property C3 (m+1, R) is replaced by C2 (m, R/N)? We shall give an affirmative answer to this question and also investigate commutativity of periodic ring satisfies C7 (m, R/N). RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

19. Theorem 1. Let R satisfies C1 (m, R/J(R)), C2 m, R/J(R)) and Q(m). Then R is a commutative ring. Theorem 2. Let R with 1 satisfies C1 ( m, R/J(R)), C7 m, R/J(R)) and Q(m). Then R is a commutative ring. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

20. In 1995[ Acta Math. Hungar.] Yaqub et al. A periodic ring R is commutative if R satisfies the property C5 (m, R/N). Also they established that if N is commutative in a periodic ring R and R is m(m+1)-torsion free ring satisfying the property C5 (m, R/N), then R is commutative. It is natural to ask a question: Do the above results valid if the property C5 (m, R/N) is replaced by C7 (m, R/N)? We settle this question and prove the following results which have been published in Mathematicum Archivum (BRNO) 2007. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

21. Theorem 3: Let m  1 be a fixed positive integer and let R be a periodic ring satisfying the properties Q (m (m+ 1)) and C7 (m, R/N). Moreover, if N is commutative, then R is commutative • In the hypothesis of Theorem 2 without identity 1 does not assure commutativity of R. Take the ring R of strictly upper triangular matrices in M3(Q), where Q is a rational number is a counter example. • We give an example to show that the property Q (m) in the hypothesis of Theorems 1 and 2 is not superfluous even if the properties C1 ( m, R/J(R)) and C2 m, R/J(R)) hold. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

22. Example. Take • Clearly, satisfies [ x3, x3] = 0 and • (x y)3 = x3 y3, for all x, y  , but does not satisfy Q(3). • GF(q) means the Galois field (or finite field) with q elements. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

23. (Contd.) Ring properties In addition , we define the properties as given below: (i) For all x,y in R, [x,y] = ±xp[ xn, ym]r xq where p ≥ 0, q ≥ 0 , n 1, r 0, m ≥ 0 are fixed non-negative integers. (ii) For all x,y in R, xs[x,y] = ±xp[ xn, ym]r xq where p ≥ 0, q ≥ 0 , m ≥ 0, s ≥ 0, r 0 n  1 are fixed non-negative integers. (iii) For all x,y in R,[x,y] xs = ±yp[ xn, ym]r yq where p ≥ 0, q ≥ 0 , n ≥ 0, s ≥ 0, r 0 m  1 are fixed non-negative integers. (iv) For each y  R,  polynomials f( g(, h( in Z( such that xp[ x,yr] yq =  g(y) [x, xs f(y)]m h(y) for all x  R and fixed integers r  1 p , q , m  1. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

24. (v) For each x, y  R,  polynomials f( • g(, h(, ğ (, ƒ(, ĥ( Z( • and r = r(x, y)>1,s = s(x, y)>1 • p = p(x, y) 0, q = q(x, y) 0, m  1 • t  2 are integers such that r and s are • relatively prime and R satisfies • xp[ xr, y] xs =  g(x) [xt f(x), y]m h(x); and • xq[ xs, y] ys =  ğ(x) [xt ƒ(x), y]m ĥ(x). • In addition , one can then include a single • remark such as“Clearly there are dual properties • obtained by replacing xp[ xr, y] yq by yq[ xr, y] xp in the definition (iv),putting xp[ xr, y] and xq[ xs, y] by [xr,y]xp and [ xs, y] xq in the definition (v)”. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

25. 5.4 OPEN PROBLEMS Question1.A ring R (may be without unity 1) is commutative if and only if R satisfies (i). Question 2. If R is a ring with unity1, then the following statements are equivalent: R satisfies (ii). R satisfies (iii). R is a commutative ring. Question 3. If R is a semi prime ring, then the following statements are equivalent: R satisfies (ii). R satisfies (iii). R is a commutative ring. Question 4. Let R be a ring with unity 1 satisfying (iv). Moreover, if R satisfies Q(r) then R is commutative (and conversely). Question 5. Let R be a left s-unital ring satisfying (v). Suppose that R satisfies (CH). Then R is commutative (and conversely). In Questions 5, one can consider R as a right s-unital rings in place of lefts-unital rings. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

26. 6. ORIGIN OF NEAR-RING Historically, the first step toward near rings was an axiomatic research done by Dickson in 1905. He showed that there do exist “fields with only one distributive law” [ i.e. named as near fields]. Later these near fields showed up again and proved to be useful in coordinatizing certain important classes of geometric planes ( recall that Descartes' method of coordinatizing the “usual” plane by the field of real numbers was one of the most successful steps in geometry). Zassenhaus (1937) was able to determine all finite near fields [i.e. named as Near-ring] . Recall, the ring structure of the endomorphism of a commutative group is well known. Most of the ring properties do not hold for the endomorphism of a non-commutative group. The system generated additively by the endomorphism and anti endomorphism of a group forms a group with respect to addition, a semi group with respect to operator multiplication, and multiplication is left distributive with respect to addition. This system is an example of near ring. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

27. A near ring is an ordered triple • N = (P, +, ∙, where P ≠Φ, “ +” addition • and “∙”multiplication are binary • composition on P such that (i) ( P, +) is a • group; (ii) (P, ∙) is a semi group; and (iii) • multiplication right distribute over addition. • A near field is a near ring in which every non zero element has a multiplicative group. • Near rings are generalized rings. • Examples • 1). Take Z= set of integers and define “∙” on Z by • x∙y = x, x, y . Then (Z, +, ∙ is a near ring. • 2). Take Z12= { 0,1,2,3,4,5,6,7,8,9,10,11} and • define “∙”on Z12 by x∙y = x, for all x, y  Z12. Then (Z12 ,+, . ) is a near ring. • If (N, +) is an abelian group, then a near ring N is an abelian near ring. • If (N, ∙) is commutative, then a near ring N is commutative near ring. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

28. 7. COMMUTATIVITY THEOREMS FOR NEAR-RINGS • It is not always easy to obtain near-ring theoretic analogue to ring theoretic results. • Neither do many of them hold in general. For example, there exists Boolean near-ring (satisfying x2 = x) which are not commutative. • Motivated by some remarks like those given by Steve Ligh [ Bull. Austral. Math. Soc.1969]. • One may reach to a conclusion that: • Many polynomial identity conditions implying commutativity in rings may turn • distributively generated (d-g) near-rings into rings. • A good deal of recent work on near-ring has been concerned with these type of results. • But the following example rules out this possibility in general. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

29. 8. EXTENSIONS OF COMMUTATIVITY OF NEAR-RINGS Theorem [Amer. Math. Monthly 1986] Searcoid and MacHale have shown that R is commutative if R satisfies the following condition: For each x, y in R, there exists an integer n = n(x, y) > 1 such that (x y)n = x y. Consider N = {0, a, b, c. u, v} with addition and multiplication, defined as follows RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

30. RECENT DEVELOPMENTS ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

31. RECENT DEVELOPMENTS ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

32. It is obvious to check that (N, +, ) is a commutative (distributive ) near ring satisfying a much simpler polynomial identity (xy)² =xy. However, N is not a commutative ring. In spite of such adverse examples are should not stop exploring the possibilities of extending the known ring theoretic results to as large a class of near rings as possible. A long standing result due to Herstein [ PAMS; 1954] asserts that a periodic ring is commutative if its nilpotent elements arecentral, and Ligh[AMM; 1989] has asked whether similar result holds for distributively generated near rings RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATIN BY MOHARRAM A. KHAN

33. In 1980 [Proc. Edinburgh Math.Soc.] Bell gave an affirmative answer and proved that: If N is a (d-g) near ring with its nilpotent elements lying in the center, then the set I of nilpotent elements forms an ideal and N/I is periodic, N must be commutative. In 1992 [Istit. Math. Univ. Trieste] Quadri et.al. proved that a (d-g) near ring satisfying the property xy = x ymx for m = m (x, y) > 1 is commutative. In 2000 [Demonstratio Math.] Khan asserts that if (d, g ) nearring N satisfies the property xy = p(x,y), where p(x,y) is a finite sum of terms of the form ix pi yqi xri, the number of summands and i , pi ,qi,ri are vary x, y and i , pi ,ri  , qi > 1. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

34. 8.1 Near –ring properties Consider N is a left near-ring, we define the following properties. (1) For each pair of x, y N, there exist positive integers p = p(x, y)  1, q = q(x, y) 1 and f(xyx) represents an element of near-ring N, which is finite sum of powers (xyx)t, t  2 with additive inverses of such powers such that xy = xp f(xyx)xq. (2) For each pair of x, y N, there exist positive integers p = p(x, y)  1 and f(xyx) represents an element of near-ring N which is finite sum of powers (xyx)t, t  2 with additive inverses of such powers such that xy = (xyx)p f(xyx). 3) For each pair of elements x, y  N, there exist integers n = n(x, y)  1 such that xy = xn ym p(x, y), where p(x,y) denotes an element of a near-ring which is the finite sum of powers of xp for p ≥ 2. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

35. In 2005[International J. Pure & Applied Math.] Khan: Let N be a (d-g) near-ring satisfying one of the conditions (1), (2), and (3).Then N is periodic and commutative. Moreover, N = M+ S, where M is a subring and S is a sub near-ring with trivial multiplication. Finally, I would like to pose questions. Before stating the problem we define the following properties. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

36. ( Contd) Near-ring properties. • 1) For each pair of elements x, y  N, there exist integers n = n(x, y)  1 • such that xy = xnym p(x, y), where p(x,y) denotes an element of a • near-ring which is the finite sum of powers of xpfor p ≥ 2. • 2) For each pair of elements x, y  N, there exist integers n = n(x, y)  1 • such that xy = yn p(x, y) ym, where p(x,y) denotes an element of a • nearing which is the finite sum of powers of xp for p ≥ 2. • For all x,y in R, either [x,y] = ±xp[ xn, ym]rxqor [x,y] = ±xp[ xn, ym]ryqwhere • p ≥ 0, q ≥ 0 , n 1, r  0, m ≥ 0 are fixed non-negative integers. • 4) For x, y N, there exist integers p = p(x, y) > 1 and q = q(x, y)>1 such that • [x,y] =  [ xn, ym]. • 5) For x, y N, there exist integers m= m(y) > 1 and fixed integers p, q such • that [x,y] = xp[ x, ym] xq. • 6) For x,yN, there exist integers n= n(x) > 1 and fixed integers p, q such • that [x,y] = xp[ xn, y] xq. • 7) There exist positive integers m, n and N admits a derivation d satisfying • either d([x, y]) = ± yp[ x, y] yn or d(xoy) = ± xp[ x, y] xn RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

37. 8.2 Open problems Question 1. Let N be a D- near-ring satisfying one of the conditions (1), (2) and (3). If idempotent elements of N are central, then N= A  B, where A is a subring and B is a sub near-ring with trivial multiplication. Question 2. Let N be a periodic (d-g) near-ring satisfying any one of the properties (4), (5), (6) and (7). Then N is a commutative ring. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

38. 8.3 Conclusion Some of the questions I have been recently interested to include the following, • What are the precise relations between the purely algebraic and the functional analytic constructions ? • To what extent is a derivation compatible with the prime ideals of non commutative rings? RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

39. 9. APPLICATIONS OF NEAR-RING . The theory of near rings is now a sophisticated theory which has found numerous applications in various areas. For instances Nearring and experimental designs Efficient code and nearring Nearring and automata Nearring and dynamical system The design of statistical system RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

40. In geometry: If two lines intersect at a unique • point, then one can define a planar nearring. • For instance: Two lines y = xa + b and y = x c +d • with a  intersect in uniquely in one point. • Since xa + b = x c + d  x a = x c + (d – b). • Definition. A nearring N is Planar (or projective) if all equations x a = x b +c, where a, b, c in N and a  b, have a unique solution. • All finite near fields are planar. • A nearring N = (N, +, .) is a near field if the set N* = { set of non-zero elements of N} forms a group with respect to multiplication. • Example. Define'.' on Z2 by x .y = x, x, y  Z2. • Then (Z2 , +, .) is a near field, where Z2 = {0, 1}. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

41. 9.1 Construction of planar near-rings There are good construction method for obtaining planar nearring see James Clay, Nearring: geneses and applications, Oxford university press, Oxford 1992]. Theorem. Let F be a field of order pn , p a prime, let t be non trivial division of pn-1so s t = pn-1 for some s. Choose a generator g of the multiplicative group of F. Define gat gb : = g a + b – [a]s where [a]s denote the residue class of a modulo s. Then N = ( F , +, ) is a planar nearring with N = N \{0}. Example. Take F be a field Z7. Then pn -1 = 6; choose t = 3 and get s= 2.This yields the planar near ring ( Z7, +7, 3. Consider Z7 = {0, 1,2, 3, 4, 5, 6} with addition “ +7” and multiplication “3” table defined as follows: RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

42. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

43. It is easy to check that ( Z7, +7, 3is a planar near ring. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

44. 10 Project 10.1 Ring properties (i) For each y in R, there exists polynomials f() , g(), h() in Z() such that xp[xr, y]xq=g(y)[x, f(y)]m h(y) for all x  R, fixed integers r > 1, p  0, q  0 and m  0. (ii) For each x, y  R, there exists polynomials f(), g(), h(), g(), h()  Z() and r = r(x, y) >1,s = s(x, y) >1,p =p(x, y) ≥ 0, q = q(x, y) ≥0, m ≥ 1, t ≥ 2 are integers with r and s are relatively prime such that xp[xr, y]= g(y)[x, ytf(y)]m h(y) and xq[xs, y]=g(y)[x, ytf(y)]mh(y). RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

45. Ring properties (iii) for any (iv) for any (v) There exists a positive integer k > 1 such that, for any (vi) For every y R there exist polynomials f(X), g(X) and h(X)Z such that either or where are fixed . RECENT DEVELOPMENT ON COMMUTATIVTY OF RINS AND ITS APPLICATION BY MOHARRAM A. KHAN

46. 10.2 Questions Question 1.A ring R (may be without unity 1)is commutative if and only if R satisfies (i). Question 2. If R is a ring with unity1, then the following statements are equivalent: R satisfies (ii). R satisfies (iii). R is a commutative ring. Question 3. Let R be a ring with unity 1 satisfying (iv). Moreover, if R satisfies Q(r) then R is commutative (and conversely). Question 4. Let R be a left s-unital ring satisfying (v). Suppose that R satisfies (CH). Then R is commutative (and conversely). Question 5. Let R be a left s-unital ring satisfying (vi). Suppose that R satisfies (CH). Then R is commutative (and conversely).Khan-UMYU_P2016.docx RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

47. 10.3 Near-rings: 10.3.1 Algebraic Properties We define the following : 1) For each pair of elements x, y  N, there exist integers n = n(x, y)  1 such that xy = xn ym p(x, y), where p(x,y) denotes an element of a near-ring which is the finite sum of powers of xp for p ≥ 2. 2) For each pair of elements x, y  N, there exist integers n = n(x, y)  1 such that xy = yn p(x, y) ym, where p(x,y) denotes an element of a nearing which is the finite sum of powers of xp for p ≥ 2. • For all x,y in R, either [x,y] = ±xp[ xn, ym]r xq or [x,y] = ±xp[ xn, ym]r yq where p ≥ 0, q ≥ 0 , n 1, r  0, m ≥ 0 are fixed non-negative integers. 4) For x, y N, there exist integers p = p(x, y) > 1 and q = q(x, y)>1 such that [x,y] =  [ xn, ym]. 5) For x, y N, there exist integers m= m(y) > 1 and fixed integers p, q such that [x,y] =  xp[ x, ym] xq. 6) For x,yN, there exist integers n= n(x) > 1 and fixed integers p, q such that [x,y] =  xp[ xn, y] xq. RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

48. 10.3.2 Differential properties If there exist positive integers m, and n such that N admits a derivation d satisfying one of the following differential properties RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

49. 10.3.3 Questions Question 1. Let N be a D- near-ring satisfying one of the conditions (1), (2) and (3). If idempotent elements of N are central, then N= A  B, where A is a subring and B is a sub near-ring with trivial multiplication. Question 2. Let N be a periodic (d-g) near-ring satisfying any one of the properties (4), (5), and (6). Then N is a commutative ring. Question 3. Let N be a prime near–ring satisfying one of the properties (i)- (v). Then N is a commutative ring. An extension on derivations in Prime Near.docx RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN

50. THANK YOU RECENT DEVELOPMENT ON COMMUTATIVITY OF RINGS AND ITS APPLICATION BY MOHARRAM A. KHAN