Noncommutativity and noncentral zero divisors

Bell, Howard E.; Klein, Abraham A.

doi:10.1155/s0161171299220674

Cited by 3 publications

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“…. , t−1, as given by (1). After cancelling terms of form f α h t−r g β that occur twice, we obtain, since f n−t g n−t does not occur twice,…”

Section: Rings With Commuting Nilpotents and Zero Divisors 81mentioning

confidence: 87%

“…Therefore R/N is commutative.Also, since C ⊆ D, C is commutative, so (i) holds; and (iii) holds sinceC 3 = {0}. Since [x 4 , x 5 ] ∈ D, x 3 [x 4 , x 5 ] ∈ D, so by (i), 0 = [[x 1 , x 2 ], x 3 [x 4 , x 5 ]] = [x 1 , x 2 , x 3 ][x 4 , x 5 ].The ring of Example 3.1 of[1] is one with D central and D 2 = {0}. We proceed to construct a ring with D commutative and noncentral, and D 2 = {0}.…”

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confidence: 99%

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Rings with Commuting Nilpotents and Zero Divisors

Klein

Bell²

2007

Result. Math.

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We explore the consequences of certain commutativity hypotheses on a single nilpotent element or the set N of all nilpotent elements. We give several sufficient conditions for N to be an ideal. We present some nontrivial examples, including an example in which N is commutative (in fact, N 2 = {0}) and N is not an ideal. Mathematics Subject Classification (2000). 16N40, 16U80.Let D = D(R) be the set of zero divisors of a ring R and N = N (R) its set of nilpotents. If N is central, and in particular if R is commutative, then N is an ideal. If N is commutative, it is a subring, but not necessarily an ideal (see Theorem 4.5). In Section 1, we study the ideals generated by the powers of a nilpotent element that commutes with all nilpotent elements. Then, in Section 2, we consider rings with N commutative and study the ideals generated by the powers of N . In Section 3, we obtain some conditions, involving larger subsets of zero divisors, under which N is an ideal. Examples of rings with N commutative such that N is not an ideal are constructed in Section 4.In [2] we studied noncommutative rings with D central. In the last section, we study rings with D commutative. Although D need not be an ideal when R is commutative, D is an ideal if it is commutative and R is not commutative. In case D 2 = {0}, R satisfies some polynomial identities; and examples of such rings will be constructed.Let D be the set of left zero divisors, D r the set of right zero divisors. Then D = D ∪ D r ; and T = D ∩ D r is the set of two-sided zero divisors. For any subset or element Y of R, let A (Y ), A r (Y ), A(Y ) and Y denote respectively, the left, right, and two-sided annihilators of Y and the ideal generated Prof.

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“…. , t−1, as given by (1). After cancelling terms of form f α h t−r g β that occur twice, we obtain, since f n−t g n−t does not occur twice,…”

Section: Rings With Commuting Nilpotents and Zero Divisors 81mentioning

confidence: 87%

mentioning

confidence: 99%