ON SOME GENERALIZATIONS OF REVERSIBLE AND SEMICOMMUTATIVE RINGS

The concept of strongly central reversible rings has been introduced in this paper. It has been shown that the class of strongly central reversible rings properly contains the class of strongly reversible rings and is properly contained in the class of central reversible rings. Various properties of the above-mentioned rings have been investigated. The concept of strongly central semicommutative rings has also been introduced and its relationships with other rings have been studied. Finally an open question raised in [D. W. Jung, N. K. Kim, Y. Lee and S. J. Ryu, Bull. Korean Math. Soc., 52(1) (2015), 247-261] has been answered. Mathematics Subject Classification (2010): 16S99, 16U80


Introduction
Throughout this article all rings are associative with identity unless otherwise stated.Let R be a ring.Denote the polynomial ring with an indeterminate x over R by R[x], the power series ring with an indeterminate x over R by R [[x]], center of R by Z(R), set of idempotent elements of R by E(R), set of nilpotent elements of R by N (R), the n by n full (resp., upper triangular) matrix ring over R by M n (R) (resp., U n (R)).Following the literature, we use D n (R) = {(a ij ) ∈ U n (R) : all diagonal entries are equal} and b st = b (s+1)(t+1) for s = 1, . . ., n − 2 and t = 2, . . ., n − 1}.Use E ij for the matrix with (i, j)-entry 1 and other entries 0; N (resp., Z n ) denotes the ring of positive integers (resp., integers modulo n), and H denotes the ring of real quaternions.
A ring R is called reduced if it has no nonzero nilpotent elements and is called central reduced [1] if all the nilpotent elements of R is central.Lambek [15] called a ring R symmetric if abc = 0 implies acb = 0 for all a, b, c ∈ R. A ring R is said to be Armendariz [22] if for any f (x) = m i=0 a i x i , g(x) = n j=0 b j x j ∈ R[x], f (x)g(x) = 0 implies a i b j = 0 for all 0 ≤ i ≤ m, 0 ≤ j ≤ n and central Armendariz [2] if for any , f (x)g(x) = 0 implies a i b j ∈ Z(R) for all i, j.A ring R is called reversible [7] if ab = 0 implies ba = 0 for all a, b ∈ R and is called central reversible [10] if ab = 0 implies ba ∈ Z(R) for all a, b ∈ R. A ring R is called semicommutative [17] if ab = 0 implies aRb = (0) for all a, b ∈ R and is called central semicommutative [20] if ab = 0 implies aRb ⊂ Z(R) for all a, b ∈ R.
The relationship among these classes of rings is given by The class of reversible rings strictly contains the class of strongly reversible rings.
Similarly a ring R is called strongly semicommutative [25] if the polynomial ring R[x] is semicommutative.Every strongly semicommutative ring is semicommutative but the converse is false.
In this paper we have defined strongly central reversible and strongly central semicommutative rings as generalizations of strongly reversible and strongly semicommutative rings respectively.We have also investigated various properties of these rings and their relationships with other known rings.Lastly an open question left unanswered in [9] has been answered.

Strongly central reversible rings
Remark 2.2.
(1) All commutative, reduced, strongly reversible rings are strongly central reversible. ( The class of strongly central reversible rings is closed under subrings and direct products. ( be the free algebra of polynomials with zero constant terms in noncommutating Note that A is a ring without identity and consider the ideal I of Z 2 + A generated by , and r 1 r 2 r 3 r 4 r 5 , where r, r 1 , r 2 , r 3 , r 4 , r 5 ∈ A. Then clearly A 5 ∈ I. Let R = (Z 2 +A)/I and consider Then by [9, Example 2.12], R is central reversible whereas Hence R is not strongly central reversible.
Example 2.4.A strongly central reversible ring need not be strongly reversible.
For a commutative reduced ring R, D ) and hence R is strongly central reversible.

ARNAB BHATTACHARJEE AND UDAY SHANKAR CHAKRABORTY
Proposition 2.9.For a ring R, the following are equivalent: (1) R is strongly central reversible. ( Proof.The equivalence of ( 1), ( 2) and ( 3) can be established by showing that R The argument here is essentially due to [3, Theorem 2]. ( =⇒ ) Let R be a strongly central reversible ring, and let where the degree is as polynomials in x and the degree of zero polynomial is taken to be 0.
and the set of coefficients of the f i 's (resp., g i 's) equals the set of coefficients of f (x k ) (resp., ) and hence Proposition 2.10.Let R be a central Armendariz ring with the property that Recall that for a ring R and an (R, R)-bimodule M , the trivial extension of R by M is the ring T (R, M ) = R ⊕ M with usual addition and the following multiplication Corollary 2.12.Let R be a reduced ring.Then T (R, R) is strongly central reversible.

ON SOME GENERALIZATIONS OF REVERSIBLE AND SEMICOMMUTATIVE RINGS 15
Proof.We have T (R, R) ∼ = R[x]/(x 2 ).The rest follows from Proposition 2.11.
We next generalize Corollary 2.12 as follows: Proposition 2.13.Let R be a central reduced ring.Then T (R, R) is strongly central reversible.
Multiplying by g 0 from left, we get Using commutativity of g 0 f 0 , we get Again multiplying by f 1 from right and using commutativity of g 0 f 0 , we get Again multiplying by g 0 from right and using f 0 g 0 = 0, we get This gives (g 0 f 1 ) 3 = 0. Therefore, g 0 f 1 is nilpotent in R[t] and so is central in From Proposition 2.13, one may suspect that if R is strongly central reversible, then T (R, R) is strongly central reversible.However the following example eradicates the possibility.
is not central in S, showing that S = T (R, R) is not central reversible and hence not strongly central reversible.
Next we shall give an example to show that a homomorphic image of a strongly reversible (hence strongly central reversible) ring need not be strongly central reversible.This may be verified as follows: Recall that an element u of a ring R is called right (resp.left) regular if ur = 0 (resp.ru = 0) implies r = 0 for r ∈ R. Proof.
The ring of Laurent polynomials in x over a ring R consisting of all formal sums n i=k r i x i with usual addition and multiplication, where Corollary 2.18.For a ring R, the following are equivalent: (1) R is strongly central reversible.

Strongly central semicommutative rings
(1) All commutative, reduced, strongly semicommutative rings are strongly central semicommutative. ( The class of strongly central semicommutative rings is closed under subrings and direct products.Proof.Let R be a central reduced ring, and let f (x), g(x As noted in [9,  (1) R λ is strongly central semicommutative for each λ ∈ Λ.
Corollary 3.9.Let R be a ring and e ∈ E(R).Then R is strongly central semicommutative if and only if both eR and (1 − e)R are strongly central semicommutative.
Proposition 3.10.For a ring R, the following are equivalent: (1) R is strongly central semicommutative. ( Proof.The equivalence of (1), ( 2) and ( 3) can be established by showing that R is strongly central semicommutative ⇐⇒ R[x] is strongly central semicommutative.
The argument here is essentially due to [3,Theorem 2]. ( =⇒ ) Let R be a strongly central semicommutative ring, and let f (t), g(t) ∈ R[x][t] with f g = 0.
We can write where the degree is as polynomials in x and the degree of zero polynomial is taken to be 0. Then f and the set of coefficients of the f i 's, g i 's and h i 's equal to the set of coefficients of f (x k ), g(x k ) and h(x k ) respectively.
Since f (t)g(t) = 0 and x commutes with elements of R, f is strongly central semicommutative.( ⇐= ) Obvious as R can be considered as a Proposition 3.11.Let R be a central Armendariz ring with the property that ) and hence R is strongly central semicommutative.
Multiplying by g 0 from left and using commutativity of g 0 f 0 , we get Again multiplying by f 1 from right and using commutativity of g 0 f 0 , we get Again multiplying by g 0 from right and using f 0 g 0 = 0, we get ON SOME GENERALIZATIONS OF REVERSIBLE AND SEMICOMMUTATIVE RINGS 21 g 0 f 1 g 0 f 1 g 0 = 0.This gives (g 0 f 1 ) 3 = 0. Therefore, g 0 f 1 is nilpotent in R[t] and so is central in R[t].
) for any positive integer n, where (x n ) is the ideal generated by x n .Proposition 2.11.Let R be a reduced ring.Then R[x]/(x n ) is strongly central reversible for any positive integer n.Proof.For any positive integer n, R[x]/(x n ) is strongly reversible by [26, Proposition 3.5] and hence R[x]/(x n ) is strongly central reversible by Remark 2.2 (1).

Example 2 . 14 .
Consider the ring R = T (H, H).By [26, Corollary 3.6], R is strongly reversible and so R is strongly central reversible

Example 2 . 15 .
Consider the ring R = D 3 (H) and the ideal I = Clearly the mapping φ : R/I → T (H, H) given by φ an isomorphism, showing that R/I ∼ = T (H, H).We know H being a division ring is reduced and therefore T (H, H) is strongly reversible by[26, Corollary 3.6] and so T (H, H) is strongly central reversible, however R is not even central reversible.

( 3 )
Every strongly central semicommutative ring R is central semicommutative, and converse holds if R is Armendariz [24, Theorem 2.21].Example 3.3.A central semicommutative ring need not be strongly central semicommutative.Let A = Z 2 [a 0 , a 1 , a 2 , a 3 , b 0 , b 1 ] be the free algebra (with identity) over Z 2 generated by six indeterminates a 0 , a 1 , a 2 , a 3 , b 0 , b 1 .Let I be the ideal of R generated by

Proposition 3 . 12 .
Let R be a reduced ring.Then R[x]/(x n ) is strongly central semicommutative for any positive integer n.Proof.For any positive integer n, R[x]/(x n ) is strongly semicommutative by [25, Example 3.9] and hence R[x]/(x n ) is strongly central semicommutative by Remark 3.2 (1).Corollary 3.13.Let R be a reduced ring.Then T (R, R) is strongly central semicommutative.We next generalize Corollary 3.13 as follows: Proposition 3.14.Let R be a central reduced ring.Then T (R, R) is strongly central semicommutative.Proof.Let u = x + (x 2 ) so that R[x]/(x 2 ) = R[u] = R + Ru, where u commutes with elements of R and u

Example 3 . 15 .
Thus, f hg ∈ Z(R[u][t]) and hence R[u] = R[x]/(x 2 ) ∼ = T (R, R) is strongly central semicommutative.From Proposition 3.14, one may suspect that if R is strongly central semicommutative then T (R, R) is strongly central semicommutative.However the following example eradicates the possibility.We consider the ring in Example 2.14, i.e., S = T (R, R), where R = T (H, H).The trivial extension R = T (H, H) strongly semicommutative by[25,    Example 3.10] and therefore by Remark 3.2 (1), R is strongly central semicommuin S, showing that S = T (R, R) is not central semicommutative and hence not strongly central semicommutative.
Proposition 2.17.A ring R is strongly central reversible if and only if ∆ −1 R is strongly central reversible.
An element is called regular if it is both right and left regular.For a ring R, we denote by ∆, a multiplicatively closed ON SOME GENERALIZATIONS OF REVERSIBLE AND SEMICOMMUTATIVE RINGS 17 subset of R consisting of central regular elements.Let ∆ −1 R be the localization of R at ∆. Then we have the following results: Lemma 2.16.For a ring R and an element x ∈ R, x ∈ Z(R) implies (x/u) ∈ Z(∆ −1 R) for all u ∈ ∆.
SOME GENERALIZATIONS OF REVERSIBLE AND SEMICOMMUTATIVE RINGS 19 3.2 (1), but R is not central reversible by [9, Example 2.6] and so it is not strongly central reversible.Proposition 3.8.Let {R λ : λ ∈ Λ} be rings.The following are equivalent: [22, Armendariz and semicommutative, and so R is strongly semicommutative by[22, Proposition 4.6] and hence R is strongly central semicommutative by Remark ON