NC-Rings and Some Commutativity Conditions

Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative rings this sum is nilpotent. In between commutative and non-commutative rings there are several types of rings in which this property holds. For instance, reduced, NI, AI (or IFP), 2-primal, reversible and symmetric, etc. We may term these types of rings as nearby commutative rings (in short NC-rings). In this work we have studied properties and various characterizations of such rings as well as rngs. As applications, we have investigated some commutativity conditions by involving semi-projective-Morita-contexts and right Ck-Goldie rings.


Introduction
This expository work deals with the rings which are near to commutative rings.
We know that sum of two nilpotent elements in a ring may not be nilpotent in general, for instance, for any ring A with 1, in M 2 (A), E 12 and E 21 are nilpotent but E 12 + E 21 is a unit.If a ring is commutative the sum of nilpotent elements is always nilpotent.In between commutative and non-commutative rings there are several types of rings in which this property holds.For instance, reduced, NI, AI (or IFP), 2-primal, reversible, and symmetric, etc.We may term these types of rings as nearby commutative rings (in short NC-rings).
In this note we will use both types of rings, we mean, associative rings that may or may not possess the multiplicative identity, and its subclass, the associative rings that possesses the multiplicative identity.For convenience, in this note, the rings of the bigger class are denoted by "r n g s" (note that "i" is missing) while that of the subclass by "r i n g s".In order to avoid unnecessary lengthy sentences we will not use the synonyms like "rings without identity" and "rings with identity", etc. Modules over rings are considered to be unital.Note that, the terminology "rngs" was used by Jacobson for the categories of rings without one in [1].
As applications, in the last section, we have investigated some commutativity conditions by involving semi-projective-Morita-contexts and right C k Goldie-rings.Though, derivations are also used and studied in several articles, we have avoided to touch them here to keep the paper of moderate length.Our focus is on NC-rings that we have studied in the second section and have discussed some properties along with examples and counter examples.We have recalled some properties of reduced, AI, NI, reversible, and symmetric rings and have proved them for rngs.It is noticed that symmetric rngs do not satisfy the initial conditions as originally defined by Lambek for rings in [2].So, they are bifurcated as left and right symmetric rngs.We have also studied those rings which are not under the class of NC-rings, but with certain conditions, they become such rings.
For instance, NCI, near-AI, quasi-AI, Armendariz, or weak-Armendariz, etc.They become NC-rings, when they are von Neumann regular.First section as usual is for preliminaries.
For basic definitions, terms and notations in ring theory we have followed the texts [1] [3].For Morita contexts, in addition to these texts we refer [4], and in particular, for projective-Morita-contexts and semi-projective-Morita-contexts one may look at [5] [6] [7].

Preliminaries
Unless otherwise stated, we assume throughout that the lower case letters , x x′ or i x are elements of the set denoted by the upper case letter X.An element a (respt.a subset X) of a rng A is called nilpotent if for some natural number n, 0 n a = (respt.0 n X = ).The least positive integer n for which 0 n a = is called the index of nilpotency.The nilpotency index of a rng A is defined to be the supremum (possibly infinite) of the nilpotency indices of all nilpotent elements of A. An ideal or a subset X A ⊆ is called nil if every element of X is nilpotent.
We will denote by ( ) N A the set of all nilpotent elements of A, by ( ) the sum of all nil ideals (the so-called upper nil radical) of A, by ( ) the intersection of all prime ideals (the so called prime or lower nil radical) of A, and by ( ) In case of rings with finite chain conditions on ideals, for instance for Noetherian and Artinian rngs, it is well known that A domain is a rng which has no non-zero left or right zero divisors and an An additive abelian group is called a zero-rng in case the product of each pair of its elements is zero.This rng is clearly commutative and its index of nilpotency is two.
A rng A is prime in case for any pair of elements, a and a′ of A, 0 aAa′ = , then either 0 a = or 0 a′ = , and semiprime if for any element a of A, 0 aAa = implies 0 a = .A is semiprime if and only if x A ∈ , such that axa a = and strongly regular (in short SR) if a A ∀ ∈ , there For any subset X of a rng A the right annihilator of X in A is a right ideal of A which is defined and denoted by: The left annihilator

( )
A l X is defined analogously and it is a left ideal of A. A right ideal I of a rng A is said to be essential if J is any other right ideal of A and if For I to be right essential, we will use the notation It is clear that a right ideal I is essential if and only if for every x A ∈ , there exists some y A ∈ such that 0 xy I ≠ ∈ .
An element a A ∈ is right singular if

( )
A r a is an essential right ideal of A.
The set of all right singular elements is denoted by ( ) . This is also a right ideal of A. The left and two-sided counter parts of these terms are defined analogously.In general, for any rng A, ( ) ( ) The rng A is called singular if ( ) If A is a ring, then A cannot be singular.Indeed, in a ring 1 A can never be a singular element.
Clearly, every reduced rng is semiprime and every commutative semiprime rng is reduced.A prime rng is also semiprime.Division rings and domains are both semiprime and reduced, while a prime rng, which is also reduced, is a domain.It is to be noted in general that ( ) N A need not be an ideal while ( ) is an ideal and a subset of ( ) A rng A is termed as 2-primal in [8] if its prime radical (the intersection of all prime ideals) is ( ) N A and SP I in [9] if for every element a of A, the factor ring ( ) Further, a rng is an NI-rng, as defined in [10] if and an NCI-rng in [11] if

( )
N A has a non-zero ideal.
Clearly, every reduced ring is NI as in this case an ideal of A.
A rng A is called an IFP rng [12] (in our term an AI-rng) in case for any pair of elements , a b of A, if 0 ab = , then 0 aAb = .Near-IFP (or Near-AI) rngs are introduced in [13] and are characterized in Proposition.1.2. ( [13]) as A is a near-AI rng if and only if for a non-zero nilpotent element a of A, AaA contains a non-zero nilpotent ideal, while almost simultaneously, but seems to be independently, quasi-IFP (in our terms quasi-AI) rngs are introduced in [14].
By definitions, every reduced rng is an AI-rng, an AI-rng is quasi-AI, a quasi-AI rng is near-AI.By a simple argument it can be deduced that the converse of above statement holds if the ring is semiprime (see [13]

( )
, K A B is said to be a "projective Morita context", in short a "pmc" (or strict), if both mc maps, , A and , B , are epimorphisms.

( )
, K A B is said to be a "semi-projective Morita context", or a "semi-pmc", if one of the mc maps, , A or , B , is an epimorphism (see [5] [6] [7] for fur- ther details).
Let A and B be rings.In case an mc ( ) , K A B of rings is a pmc, i.e., if both mc maps , A and , B are epimorphisms, then they become isomorphisms.In this case, the category of right (respt.left) A-modules is equivalent to the category of right (respt.left) B-modules i.e., ( ) and moreover, ( ) ( ).

NC-Rngs and Their Relations
In this section we will introduce nearby commutative rngs (in short NC-rngs).
We will investigate some properties of NC-rngs and demonstrate some examples and counter examples.Then we will discuss and expose some of those rngs which satisfy the condition of NC-rngs.For instance, reduced, AI, reversible and symmetric rngs.
For rings following sequence is given in ( [9]; Fig. 1, Various types of 2-primal rings): In fact, symmetric rings are reversible, but a reversible ring may not be symmetric [9] [15].In case of rngs a symmetric rng may not be reversible also (see Example 3.4.7 ( 2)).Note that, the definition of a symmetric ring, as introduced by Lambek [2], does not justify its property for a symmetric rng.Thus we need to bifurcate it into two parts, right symmetric rng and left symmetric rng (see [16] for details).We have given examples that they are different rngs.They become identical at least if the rng is commutative or if it is a ring.Thus we have the following extended sequences for rngs which are irreversible in general (see counter examples in [9] [10] [15] [16]).

NC-Rngs
Definition 3.1.1.A rng A is nearby commutative, in short an NC-rng, in case the sum of any two nilpotent elements of A is nilpotent.Proposition 3.

A rng A is NC if and only if N(A) is an additive subgroup of A.
Proof: One way is obvious.Assume that A is an NC-rng.Clearly, ( ) ( ) , 2 are nilpotent but the sum .This is a rng, called the Klein 4-rng or right absorbing ring in [16].Its only non-zero nilpotent element is a + b and the only nilpotent ideal is {0, a + b}.Hence V is an NC-rng as well as an assume that the relations in V also hold in R. Again, {0, a + b} is the only nilpotent ideal of R. Hence R is an NC-ring as well as an NI-ring.
Example 3.1.6.A ring which is an NC-ring but not an NI-ring.
Let G be a free semigroup generated by { } : i x i I ∈ and R any NI-ring.Then the group ring RG is the free R-ring, : Let the set of nilpotent elements be ( ) N A is ad- ditively and multiplicatively closed, ( ) Let n be the index of nilpotency of y.Then,  a A ∀ ∈ is nilpotent of index two.Above arguments also show that property of being reduced is not Morita invariant.

Reduced Rings
In the following proposition the results in literature are available for rings.We verify that these are also hold for rngs.
1) Reduced rngs are semiprime and a commutative semiprime rng is reduced.
3) Reduced rngs are nonsingular and a commutative nonsingular rng is reduced. 4)If A is a nonsingular rng and

( ) ( )
N A Cent A ⊆ , then A is reduced.
Proof: 1) Obviously, reduced rings have no non-zero nilpotent ideals, so they are semiprime.Conversely, assume that 2) Indeed, if A is an SR rng and a A ∈ is nilpotent with index n, then there exists x A ∈ such that Hence A is reduced.
4) First we prove that every nilpotent element that lies in the center is singular.Let a A ∈ be a nilpotent element of index n that lies in the center of A. Then ( ) Because a is a central element, ( ) . This means that ( ) . By similar arguments we conclude that ( ) and so ( ) x N A Cent A ∈ ⊆ then x must be singular.But A is nonsingular, then x must be 0. Hence A is reduced.

AI-Rngs and Generalizations
AI-rngs have many names in literature.H.E. Bell in [12] called an ideal I of A to be with the insertion-of-factors-property in case for any pair of elements , a a′ of A, aa I ′ ∈ implies that aAa I ′ ⊆ .Thus any rng with the property that for 0 aa′ = ⇒ 0 aAa′ = is popular as an IFP-rng, the short form of insertion of factors property.Marks in [9] called it with property (S I).He continued this term from Shin in [17].Narbonne called it semicommutative (see reference in [13]) and many others followed this term (e.g., [18] [19]).Habeb in [20] called it zero insertive.We prefer to call it an AI-rng due to the fact that in this rng all "Annihilators", left or right, are "Ideals".Lemma 3.3.1.For any rng A the following are equivalent: 1) For any pair , 2) Every right annihilator ( ) A r X of X A ⊆ is an ideal.
3) Every left annihilator ( ) A l X of X A ⊆ is an ideal. 4)For every a A ∈ , right annihilator ( ) A r a is an ideal. 5)For every a A ∈ , left annihilator ( ) A l a is an ideal.

( ) ( )
A A l a l ba ≤ .
Proof: 1) ⇔ 2) Let S ≠ Φ be a subset of A. Then

( )
A r S is a right ideal of A. We prove that if 1) holds, then it is also a left ideal of A. Let The rest can analogously be proved. Definition 3.3.2.If any one of the conditions of Lemma 3.3.1 is satisfied then the rng is called an AI-rng.If 1 A ∈ , then it is an AI-ring.
It is clear from the sequences 3) that AI is an NC-rng.We pose here an alternate proof.
Proposition 3.3.3.AI-rng is an NC-rng.Proof: Let A be an AI-rng and , a b A ∈ be two nilpotent elements of indices , m n , respectively.We want to prove that a b + is nilpotent.A monomial in the expansion of ( ) t a b + is an expression of the form ( ) , where max , .

= ⇒ = ⇒ =
Such insertions eventually make 0 x = and so we conclude that ( ) Proof: Let A be a semiprime AI-rng and 0 x A ≠ ∈ be such that Then there exists 0 a A ≠ ∈ , such that ( ) Hence A is nonsingular. Marks in [9] [10] [15] used the term NI for those rings in which ( ) N A is an ideal, then naturally,

( ) ( ) N A N A * =
. A ring A is called left (right) duo if each left (right) ideal is an ideal.Proposition 3.3.5.Following are true for any rng. 1) Every left (right) duo rng is an AI-rng.
2) Every reduced rng is an AI-rng.
3) Every AI-rng is an NI-rng.4) Every idempotent in an AI-domain is central.Hence an AI-domain is abelian.
5) Every idempotent in an AI-ring is central.Hence an AI-ring is abelian.6) For any rng A, Proof: 1) Let A be a left duo ring.Let for some , a b A ∈ , ( ) ( ) r A ∀ ∈ .Hence A is an AI-rng.
2) Let A be a reduced ring.Assume that for some pair of elements , a b A ∈ , 0 ab = .Then for any x A ∈ ,  In particular, if A is an AI-ring, then 3 UT is not an AI ring. (Simply, replace a by 1 in above example).Hence ( ) n M A is not an AI-ring.This shows that for any ring being AI is not Morita invariant.
is an AI-rng.Hence, S is an NC-rng.b) If A is an AI-ring, then S is an AI-ring.Proof: a) For the verification of this claim, assume that 1 2 0 s s = .Then we get the following identities 1) 1 2 0 a a = .
4) 1 223 123 2 0 a a a a + = .Now we claim that the product

s Ss =
For this it is enough to prove that 5) 1 2 0 a Aa = .
Hence R is not an AI-ring.
Definitions 3.3.9.Let A be a rng.We define that: 1) A is an NCI-rng in case ( ) For detailed studies of these classes of rngs we refer the reader to the above cited articles.It is clear by definitions that Then R is neither abelian nor AI, but it is near-AI.
3) Now we pose an example for near-AI quasi-AI ⇐ / .
Let R be a ring and 0 I ≠ a nilpotent ideal of R such that every element of R\I is a unit.(For example a local ring).By Prop.1.10 in [13], ( ) and nilpotent.Hence we conclude that it is not quasi-AI.
If a ring A is of bounded index of nilpotency, then A is reduced if and only if A is NI (or NCI) and is semiprime ( [11]; Prop.1.3).

Reversible and Symmetric Rngs
Q Z is a reversible ring.In fact, the elements of ( ) Q Z is reversible.Moreover, from Proposition 6 of [15] it is clear that ( ) Remarks 3.4.6. 1) If the ring A is equipped with identity, say 1 R , then Lambek proved that: A is a symmetric ring if and only if for any i a A ∈ , where 1, , i n =  , the product ( ) is a permutation.This characterization may not hold if A does not have 1 A .
2) If a rng A satisfies the condition ( ) then in [23] it is called a ZC n -rng.It is proved there that for all 3 n ≥ , if A satisfies ZC n then A also satisfies ZC n+1 .Hence, inductively, ZC 3 implies ZC n for all 3 n ≥ ( [23]; Theorem I.1 & Corollary I.2).The converse does not hold in gener- al.Moreover, ZC 2 does not imply ZC 3 ([23] 0 E E = . 3) In Example 3.4.2(6) in ( ) V 3 is right symmetric but not left symmetric.On the other hand its opposite rng (V 3 ) op is left symmetric and not right symmetric.
3) By iteration, V n and (V n ) op rngs can be constructed which are right and left symmetric rngs, respectively, but they are not symmetric.For details see [16], where these are termed as generalized Klein-4 rngs or right absorbing rings.These are zero-divisor rings, and it is proved in [24] that their zero-divisor graphs are precisely the union of a complete graph and a complete bipartite graph.
1) For any i a A ∈ , where 1, , i n =  , the product ( ) It is also clear that if A is a ring, then left symmetric symmetric right symmetric.⇔ ⇔ Proposition 3.4.12.For any rng A, the following hold.
1) Every reduced rng is left and right symmetric, hence it is symmetric.
2) Every left (or right) symmetric rng is an AI-rng.
3) A left (or right) symmetric and reversible rng is symmetric. 4)Every symmetric ring is reversible.5) Every symmetric rng is an NC-ring. Proof: 1) Let A be a reduced rng.Assume that for some , , a b c A ∈ , 0 abc = .Be- cause every reduced rng is an AI-rng, so ( ) ( )( ) Similarly, 0 bac = .
2) Let A be a left symmetric rng.Assume that for some , a b A ∈ , 0 ab = .
Then r A ∀ ∈ , 0 0 rab arb =⇒ =.Hence A is an AI-rng.The rest of the proof is similar.
3) Let A be a right symmetric and reversible rng.Assume that some , , a b c A ∈ , 4) This holds because of the multiplicative identity.
5) Assume A is a symmetric rng and that for some exponents x,y, 0 x a = and 0 y b = .Then 0 x y a b = .Hence by Proposition 3.4.12,any binomial coeffi- cient in the form: , where 2) Follows from ([11]; Proposition 1.4).

Armendariz Rings and Generalizations
Finally, we very briefly review Armendariz rings.
A is called weak Armendariz in [31] in case Proof: For: Reduce rings ⇒ Armendariz rings see ([13]; Lemma 1.1).Armendariz rings ⇒ weak Armendariz rings holds by definitions.For: weak Armendariz rings ⇒ abelian rings also see ([13]; Lemma 1.1). Example 4.3.3.Armendariz ⇒ / Reduced: It is clear from above that every reduced rng is Armendariz but the converse is not true in general as it clear from the following example: For any reduced ring A, the ring of Example 3.3.7.
is also abelian (see details in [21]).Then consider the polynomials:  Remark: For vNR rings above results can also be followed from [9]

Conclusion
This expository work deals with the rings in which sum of two nilpotent elements is nilpotent.All commutative rings have this property, so we have termed them near commutative, or in short, NC-rings.In general, we have considered rings not necessarily be with one.In this work we have picked very common classes of rings which can be subsumed under NC-rings.In any future work more classes of rings can be studied and compared with NC-rings.
right and left annihilators of a, respectively.
Field, division rings, domains, reduced, symmetric, reversible, 2-primal, AI, and NI-rngs are NC-rngs.The smallest class of non-commutative NC-rngs is the class of reduced rngs.Conversely, Example 3.1.6below shows that an NC-rng properly subsumes all above mentioned classes of rngs.Hence, extending sequences (2) we have following irreversible sequences of rngs: ⊆ Right Symmetric ⊆ S. K. Nauman, N. M. Muthana DOI: 10.4236/apm.2019.92008148 Advances in Pure Mathematics Reduced ⊆ Symmetric ⊆ Reversible ⊆ AI ⊆ PS I ⊆ 2-Primal ⊆ NI ⊆ NC-rng (3) ⊆ Left Symmetric ⊆ Proposition 3.1.4.Being "NC-rings" is not a Morita invariant property.Proof: For any ring A, it is known that A and ( ) n M A are Morita similar.Let A be an NC-ring.But ( ) n M A is not an NC ring.Because, 1 1

1 )
for a ring, being NC is not Morita invariant. Example 3.1.5.(NC-rngs which are also NI-rngs).Consider the non-commutative ring of order 4, See also Remarks 3.4.8. and Examples 3.4.10

6 )
some natural number t n m ≥ + .So ( ) N R is an additive group.Hence ( ) N A is an ideal.4) Let A be an AI-domain (recall that a domain is a rng without non-zero zero-divisors).Let e A ∈ be an idempotent.Then r A ∀ ∈ , Holds by definitions. S. K.

Definition 3 . 4 . 1 . 6 )
Cohn in[22] called a ring A to be reversible in case whenever 0 ∀ ∈ .Anderson and Camillo in[23] called it a ZC2-rng.Examples 3.4.2: 1) All commutative rngs, domains, and zero-rngs are reversible.S. K. Nauman, N. M. Muthana DOI: 10.4236/apm.2019.92008155 Advances in Pure Mathematics2) Reduced rngs are also reversible.Indeed, if 0 ab = , then is not reversible.Hence the property of being reversible is not Morita invariant.4) If A is a reversible rng, then the polynomial rng [ ] A x may not be reversi- ble (see ([18]; Example 2) & ([19]; Example 2.1)).5) Example 3.1.5.(1) Clearly V is not reduced as a + b is non-zero nilpotent element and it is not reversible as Similarly the ring in Example 3.1.5.2) is neither reduced nor reversible.Consider the group of quaternions

8 Q Z is right duo. Proposition 3 . 4 . 3 . 2 )
Reversible rngs are1) AI-rngs and 2) NC-rngs.Proof: 1) Let A be reversible.Then for some , Because every AI-rng is NC, so is a reversible rng. Proposition 3.4.4.A reversible vNR ring is an SR ring.Proof: Let A be a vNR ring.Then a A ∀ ∈ , there exists r A ∈ Hence, A is SR. Definition 3.4.5.J. Lambek in[2] introduced symmetric rngs: A rng A is symmetric if for any , ,

8 . 2 )
Notice that, in the rng, with 0 0 abc acb =⇔ =, there is no guaranty that 0 bac = .For instance, consider the rng V of Example 3.1.5(1).Let c = a + b.One can easily see that abc = 0 but cab ≠ 0. This observation and Remark 3.4.6(2) suggest a split definition for a symmetric ring.Thus we define: Definitions 3.4.9.[16] A rng A is called right symmetric if for any triple , , If a rng is both left and right symmetric, it is a symmetric rng.Examples 3.4.10. 1) The rng V = V 2 as constructed in Example 3.1.5(1) or in Remarks 3.4.8, is right symmetric but not left symmetric.Obviously, the opposite rng (V 2 ) op is left symmetric but not right symmetric.Consider another example which is an extension of above example.Let be with three generators and of characteristic 2. Minor computations show that
. Straightforward calculations show that it is Armendariz.Example 4.3.4.([13]; Example 1.2) weak Armendariz ⇒ / Armendariz.Consider the factor ring: x y y = Z It is proved in ([31]; Example 3.2) that A is weak Armendariz.Now, in the polynomial ring not Armendariz.Example 4.3.5.(An abelian ring which is not weak Armendariz) Let A be an abelian ring

A and , B are the Morita maps (in short, mc maps). The images
where , a pmc of rings, then the rings A and B are said to be Morita similar (or Morita equivalent).Common properties shared by Morita similar rings are termed as Morita invariant.For instance, being prime or semiprime are Morita invariant, while being reduced, commutative, domain, division rings or fields are not Morita invariant.
RG is neither NI nor NCI, but it is clear that sum of two nilpotent elements in RG is nilpotent, so it is an NC-ring.Let x and y be nilpotent elements of the ring A. Then the group ring RG is the free R-ring : N R is not an ideal in RG.Hence Proposition 3.1.7.In a ring A, if ( ) U A is abelian, then A is an NC-ring.Proof: + + = . Examples 3.1.8.The converse of above theorem does not hold in general.For instance, in the integral ring of quaternions, ( ) = or that index of nilpotency of A is one.Because a reduced ring has no non-trivial nilpotent elements, it is an NC-rng.
Definition 3.2.1.A rng A is reduced if it has no non-trivial nilpotent element, equivalently ( ) 0 N A 2 UTM A the matrix aE 12 , Examples 3.3.6. 1) Fields, division rings, integral domains are AI-rings and all commutative rngs, domains, zero-rngs, and reduced rngs are AI-rngs.2) Being AI is not Morita invariant.
Nauman, N. M. Muthana DOI: 10.4236/apm.2019.92008153 Advances in Pure Mathematics [27] not weak Armendariz.Now we give an extended list of equivalent von Neumann regular rings.For proof we refer to ([27]; Lemma).All these rings are NC-rings.Theorem 4.3.6.Let A be a vNR ring.Then the following are equivalent. Hence The consequences of the Theorems 4.2.3 and 4.3.6 are the following.Corollary 4.2.7.A C k -vNR ring is commutative if any one of the properties (1)-(17) of Theorem 4.3.6 is satisfied.Corollary 4.2.8.A semiprime right Goldie C k -ring is commutative if its clas-sical ring of quotient satisfies any one of the properties (1)-(17) listed in Theorem 4.3.6. ∎