Reflexivity of Rings via Nilpotent Elements

An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\in$ nil$(R)$, $b\in R$, being $aRb \subseteq I$ implies $bRa \subseteq I$ where nil$(R)$ is the set of all nilpotent elements of $R$. The ring $R$ is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal $I$ of a ring $R$, $R/I$ is left N-reflexive. If an ideal $I$ of a ring $R$ is reduced as a ring without identity and $R/I$ is left N-reflexive, then $R$ is left N-reflexive. If $R$ is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in $R[x]$ are nilpotent in $R$, it is proved that $R$ is left N-reflexive if and only if $R[x]$ is left N-reflexive. We show that the concept of N-reflexivity is weaker than that of reflexivity and stronger than that of left N-right idempotent reflexivity and right idempotent reflexivity which are introduced in Section 5.


Introduction
Throughout this paper, all rings are associative with identity. A ring is called reduced if it has no nonzero nilpotent elements. A weaker condition than reduced is defined by Lambek in [16]. A ring R is said to be symmetric if for any a, b, c ∈ R, abc = 0 implies acb = 0. Equivalently, abc = 0 implies bac = 0. It is easily checked that if R is a reduced ring, then the following condition holds: ab = 0 implies ba = 0 for any a, b ∈ R. Cohn [7] called a ring R reversible if this condition holds. Anderson and Camillo [3] studied the rings whose zero products commute, and used the term ZC2 for what is called reversible. Prior to Cohn's work, reversible rings were studied under the names of completely reflexive and zero commutative by Mason [17] and Habe [8], respectively. Tuganbaev [18] investigated reversible rings under the name of commutative at zero. It is obvious that commutative rings and reduced rings are reversible. The reversible property of a ring is generalized as: A ring R is said to satisfy the commutativity of nilpotent elements at zero ([2, Definition 2.1]) if ab = 0 for any a, b ∈ nil(R) implies ba = 0; for simplicity, such a ring is called CNZ.
In [17], a right ideal I of R is said to be reflexive if aRb ⊆ I implies bRa ⊆ I for any a, b ∈ R. A ring R is called reflexive if 0 is a reflexive ideal of R. Reversible rings are reflexive by [14,Proposition 2.2]. In [19], R is said to be a weakly reflexive ring if aRb = 0 implies bRa ⊆ nil(R) for any a, b ∈ R. In [13], a ring R is said to be nil-reflexive if aRb ⊆ nil(R) implies that bRa ⊆ nil(R) for any a, b ∈ R. In [1], R is called a reflexivity with maximal ideal axis ring (an RM ring, for short) if for a maximal ideal M and for any a, b ∈ R, aM b = 0 implies bM a = 0; similarly, R has reflexivity with maximal ideal axis on idempotents (simply, RMI ) if eM f = 0 for any idempotents e, f and a maximal ideal of M yields f M e = 0. In [15], R has reflexive-idempotents-property (simply, RIP) if eRf = 0 for any idempotents e, f yields f Re = 0. A left ideal I is called idempotent reflexive [11] if aRe ⊆ I implies eRa ⊆ I for a, e 2 = e ∈ R. A ring R is called idempotent reflexive if 0 is an idempotent reflexive ideal. Kim and Baik [12] introduced the left and right idempotent reflexive rings. A two sided ideal I of a ring R is called right idempotent reflexive if aRe ⊆ I implies eRa ⊆ I for any a, e 2 = e ∈ R. A ring R is called right idempotent reflexive if 0 is a right idempotent reflexive ideal. Left idempotent reflexive ideals and rings are defined similarly. If a ring R is left and right idempotent reflexive, then it is called an idempotent reflexive ring.
Kheradmand et al. [10] generalized the notion of reflexive rings to RNP rings. A ring R is called RNP (reflexive-nilpotents-property) if aRb = 0 for any a, b ∈ nil(R) implies bRa = 0. In this paper, motivated by these classes of types of reflexive rings, we introduce left N-reflexive rings and right N-reflexive rings. We prove that some results of reflexive rings can be extended to the left N-reflexive rings for this general setting. We investigate characterizations of left N-reflexive rings and many families of left N-reflexive rings are presented. The concept of one-sided N-reflexivity for rings is placed between reflexive rings and RNP rings.
In what follows, Z denotes the ring of integers and for a positive integer n, Z n is the ring of integers modulo n. We write M n (R) for the ring of all n × n matrices; U (R), nil(R) will denote respectively the group of units and the set of all nilpotent elements of R; U n (R) is the ring of upper triangular matrices over R for a positive integer n ≥ 2; D n (R) is the ring of all matrices in U n (R) having main diagonal entries equal; and V n (R) is the subring of U n (R) described , where e i,j is the matrix unit having 1 in the (i, j) entry and 0 elsewhere, . . , n − 1}, and V n (R) = RI n + RV + · · · + RV n−1 for a positive integer n.

N-reflexivity of rings
In this section, we introduce some classes of rings, so-called left N-reflexive rings and right N-reflexive rings. These classes of rings generalize reflexive rings. We investigate which properties of reflexive rings hold for the left N-reflexive rings and right N-reflexive rings. We supply an example to show that there are left N-reflexive rings that are neither right N-reflexive nor reflexive nor reversible. It is shown that the class of left N-reflexive rings is closed under finite direct sums. We have an example to show that homomorphic images of a left N-reflexive ring need not be left N-reflexive. Then we determine under what conditions a homomorphic image of a ring is left N-reflexive. We now give our main definition. Definition 2.1. Let R be a ring. An ideal I of R is called left N-reflexive if for any a ∈ nil(R) and b ∈ R, aRb ⊆ I implies bRa ⊆ I. The ring R is called left N-reflexive if the zero ideal is left N-reflexive. Similarly, I is called right N-reflexive if for any a ∈ nil(R) and b ∈ R, bRa ⊆ I implies aRb ⊆ I. The ring R is called right N-reflexive if the zero ideal is right N-reflexive. The ring R is called N-reflexive if it is both left and right N-reflexive.
Every reflexive ring and every semiprime ring is N-reflexive. There are left Nreflexive rings which are neither semiprime nor reduced nor reversible. The concept of one-sided N-reflexivity for rings is placed between reflexive rings and RNP rings:  Proof. Let F be a field. Then nil( Then CD = 0 and DC = 0.
Hence R is not left N-reflexive.
Let F be a field and R = F [x] the polynomial ring over F with x an indeterminate. Let α : R → R be a homomorphism defined by α(f (x)) = f (0), where f (0) is the constant term of f (x). Let D α 2 (R) denote the skewtrivial extension of R by R and α. So D α ∈ R is a ring with componentwise addition of matrices and multiplication as follows: There are left N-reflexive rings which are neither reflexive nor semiprime. The N-reflexive property of rings is not left-right symmetric.

Example 2.3.
Let D α 2 (R) denote the skewtrivial extension of R by R and α as mentioned above. Then by [19, Example 3.5 Proof. Let eae ∈ eRe be nilpotent and ebe ∈ eRe an arbitrary element with eaeRebe = 0. Then we have ebeReae = 0, since R is left N-reflexive.
For any positive integer n, the full matrix ring M n (F ) over any field F is Nreflexive but M n (F ) has some subrings neither left N-reflexive nor right N-reflexive, as shown below.
(2) Let F be a field and consider the subrings U n (F ) and D n (F ) of M n (F ). It is obvious that these subrings are neither left N-reflexive nor right N-reflexive.
There are some subrings of M n (R) that are N-reflexive. Proposition 2.6. Let R be a commutative ring. Then V n (R) is an N-reflexive ring.
The commutativity of the ring R in Proposition 2.6 is not superfluous. Example 2.7. Let R be a ring and consider the ring S = V 2 (U 2 (R)). Note that U 2 (R) is not commutative. Let A = e 12 + e 14 + e 34 ∈ nil(S) and B = e 11 + e 12 + e 33 + e 34 ∈ S, where e i,j is the matrix unit having 1 in the (i, j) entry and 0 elsewhere. Then ASB = 0 and BA = 0. Hence S is not left N-reflexive.

Lemma 2.8. N-reflexivity of rings is preserved under isomorphisms.
Theorem 2.9. Let R be a ring and n a positive integer.
Proof. Suppose that M n (R) is a left N-reflexive ring. Let e ij denote the matrix unit whose (i, j) entry is 1 and whose other entries are 0. Then R ∼ = Re 11 = e 11 M n (R)e 11 is N-reflexive by Proposition 2.4 and Lemma 2.8.

Proposition 2.10. Every reversible ring is N-reflexive.
The converse statement of Proposition 2.10 may not be true in general as shown below.  For any element a ∈ R, r R (a) = {b ∈ R | ab = 0} is called the right annihilator of a in R. The left annihilator of a in R is defined similarly and denoted by l R (a).

Proposition 2.13. Let R be a ring. Then the following hold:
( Proof. (1) For the necessity, let x ∈ r R (aR) for any nilpotent element a ∈ R. We have (aR)x = 0. The ring R being left N-reflexive implies xRa = 0. So x ∈ l R (Ra). For the sufficiency, let a ∈ nil(R) and b ∈ R with aRb = 0. Then b ∈ r R (aR). By hypothesis, b ∈ l R (Ra), and so bRa = 0. Thus R is left N-reflexive.
For a field F , D 3 (F ) is neither left N-reflexive nor right N-reflexive. But there are some subrings of D 3 (F ) that are N-reflexive as shown below. Proposition 2.14. Let R be a reduced ring. Then the following hold: The condition of R being reduced in Proposition 2.14 is not superfluous, as the following example shows. Let R be a ring and I an ideal of R. Recall (see [6]) that I is called idealsymmetric if ABC ⊆ I implies ACB ⊆ I for any ideals A, B, C of R. In this vein, we mention the following result. Let R be a ring and I an ideal of R. In the short exact sequence 0 → I → R → R/I → 0, I being N-reflexive (as a ring without identity) and R/I being N-reflexive need not imply that R is N-reflexive. (1) R is a semiprime ring.
(3) R is a reflexive ring.
(4) R is a left N-reflexive ring.
Proof. Similar to the proof of Theorem 2.21.
Question: If a ring R is N-reflexive, then is R a 2-primal ring?
There is a 2-primal ring which is not N-reflexive.
Let R be a ring and S the subset of R consisting of identity and central regular elements. Set S −1 R = {s −1 r | s ∈ S, r ∈ R}. Then S −1 R is a ring with an identity.

Corollary 3.3. For a ring R, R[x] is left N-reflexive if and only if
. Then S consists of 1 and central regular elements. So the claim holds by Theorem 3.2.  (2) Consider the ring

Proposition 3.4. For a ring R, R[x] is left N-reflexive if and only if
, we have f (x)Rg(x) = 0, and so by [  Note that in the commutative case, the coefficients of any nilpotent polynomial are nilpotent. However, this is not the case for noncommutative rings in general. Therefore in Proposition 3.6 the assumption "coefficients of any nilpotent polynomial in R[x] are nilpotent in R" is not superfluous, as the following example shows.

Applications
In this section, we study some subrings of full matrix rings whether or not they are left or right N-reflexive rings.  (1) If R is a reduced ring, then H (0,0) (R) is N-reflexive but not reduced.

Generalizations and some examples
In this section, we introduce left N-right idempotent reflexive rings and right N-left idempotent reflexive rings, to generalize the reflexive idempotent rings in Kwak and Lee [14], Kim [11], and Kim and Baik [12]. We introduce the following classes of rings to produce counterexamples related to left N-reflexive rings. These classes of rings will be studied in detail in a subsequent paper by the authors. (1) Let F be a field and A = F X, Y denote the free algebra generated by noncommuting indeterminates X and Y over F . Let I denote the ideal generated by Y X. Let R = A/I and let x = X +I, y = Y +I ∈ R. It is proved in [11,Example 5] that R is abelian and so R has reflexive-idempotents-property but not reflexive by showing that xRy = 0 and yRx = 0. Moreover, xyRx = 0 and xRxy = 0. This also shows that R is not left N-reflexive since xy is nilpotent in R.
(2) Let F be a field and let A = F X, Y denote the free algebra generated by noncommuting indeterminates X and Y over F . Let I denote the ideal generated by X 3 , Y 3 , XY , Y X 2 , Y 2 X in A. Let R = A/I and let x = X + I, y = Y + I ∈ R. Then in R, x 3 = 0, y 3 = 0, xy = 0, yx 2 = 0, y 2 x = 0. In [1, Example 2.3], xRy = 0, yRx = 0, and idempotents in R are 0 and 1. Hence for any r ∈ nil(R) and e 2 = e ∈ R, rRe = 0 implies eRr = 0. Thus R is left N-right idempotent reflexive. We show that R is not a left N-reflexive ring. Since any r ∈ R has the form r = k 0 + k 1 x + k 2 x 2 + k 3 y + k 4 y 2 + k 5 yx and x is nilpotent, as noted above, xRy = 0. However, yRx = 0 since yx = 0. Thus R is not left N-reflexive.
(3) Let F be a field of characteristic zero and A = F X, Y, Z denote the free algebra generated by noncommuting indeterminates X, Y , and Z over F . Let I denote the ideal generated by XAY and X 2 − X. Let R = A/I and let x = X + I, y = Y + I, z = Z + I ∈ R. Then in R, xRy = 0 and x 2 = x. So xy = 0, yx is nilpotent, x is idempotent, and xRyx = 0. But yxRx = 0. Hence R is not right N-left idempotent reflexive. In [14,Example 3.3], it is shown that R is right idempotent reflexive.