On the commutativity of certain rings

Forsythe, Alexandra; McCoy, Neal H.

doi:10.1090/s0002-9904-1946-08603-6

Cited by 34 publications

(33 citation statements)

References 7 publications

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“…We first formally establish the result mentioned in the opening sentence of the Introduction; special cases of this have previously been noted b y Forsythe and McCoy ( [7], Lemma 1) and Herstein ([8], Lemma 4). THEOREM …”

Section: General Resultsmentioning

confidence: 65%

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Rings with Central Idempotent or Nilpotent Elements

Drazin

1958

Proceedings of the Edinburgh Mathematical Society

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Introduction.It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of 7r-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be w-regular).We discuss particularly the special case of a ring R (possibly with operators) having minimal condition on (say) left ideals; such an R is necessarily 77-regular (see [1]). It is well known that, if a given left ideal A of R contains no non-zero idempotent, then A must be nilpotent; and of course the converse of this is obvious (in any ring whatever). We consider in our concluding section what can be said along these lines if we replace the nilpotency of A by the weaker condition that R contains a non-zero ^4-annihilator. For any non-zero left ideal A whose idempotent elements are all central (in A) we obtain the following analogue: if A contains no element acting as a two-sided identity on A, then A contains a two-sided A-annihilator (the converse again being trivial). This leads at once to some simple sufficient conditions for the existence of one-sided" A -annihilators.Our arguments throughout are of a very straightforward and elementary nature, but the results do nevertheless seem worth putting on record. For brevity, we shall call a given associative ring (or algebra) R a CN-ring whenever every nilpotent element of R is central, and a CI-ring whenever every idempotent element of R is central; when we speak of A as being a Cl-ideal of R, we shall mean that A is an ideal of R and that, considered as a ring in its own right, A is itself a C/-ring. Finally, given any two elements u, v of a ring, we shall denote their additive commutator uv-vu by [u, v\.

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Section: General Resultsmentioning

confidence: 65%

“…Our last result in this section, which dates back as far as Peirce (cf. also [7], Lemma 2 and [8], Lemma 9), is only mildly relevant to the subject in hand, but we include it now for completeness; we shall call an idempotent element e of a ring R trivial (in R) if either e = 0 or e acts as a two-sided identity on R. THEOREM …”

Section: Every Cn-ring Is a Cl-ringmentioning

confidence: 99%

Rings with Central Idempotent or Nilpotent Elements

Drazin

1958

Proceedings of the Edinburgh Mathematical Society

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“…These statements are summarized in the next theorem. If A is a semi-simple ring, then any ideal of A is semi-simple and therefore (8) contains no nonzero left annihilator. Thus Lemma 6 shows directly that in the semi-simple case condition Cr implies condition C. In fact, in this last statement condition Cr may be replaced by the following condition : every ideal of A has a right ideal complement.…”

Section: Theoremmentioning

confidence: 99%

Ideal lattices and the structure of rings

Blair¹

1953

Trans. Amer. Math. Soc.

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It is well known that the set of all ideals(2) of a ring forms a complete modular lattice with respect to set inclusion. The same is true of the set of all right ideals. Our purpose in this paper is to consider the consequences of imposing certain additional restrictions on these ideal lattices. In particular, we discuss the case in which one or both of these lattices is complemented, and the case in which one or both is distributive.In §1 two strictly latticetheoretic results are noted for the sake of their application to the complemented case. In §2 rings which have a complemented ideal lattice are considered. Such rings are characterized as discrete direct sums of simple rings. The structure space of primitive ideals of such rings is also discussed. In §3 corresponding results are obtained for rings whose lattice of right ideals is complemented.In particular, it is shown that a ring has a complemented right ideal lattice if and only if it is isomorphic with a discrete direct sum of quasi-simple rings. The socle [7](3) and the maximal regular ideal [5] are discussed in connection with such rings. The effect of an identity element is considered in §4. In §5 rings with distributive ideal lattices are considered and still another variant of regularity [20] is introduced. It is shown that a semi-simple ring with a distributive right ideal lattice is isomorphic with a subdirect sum of division rings. In the concluding section a type of ideal, introduced by L. Fuchs [9] in connection with commutative rings with distributive ideal lattice, and which we call strongly irreducible, is considered. Some properties of these ideals, analogous to corresponding ones for prime ideals [19], are developed. Finally, it is observed that a topology may be introduced in the set of all proper strongly irreducible ideals in such a way that the resulting space contains the spaces of prime [19] and primitive [13] ideals as subspaces.

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“…Recall that a ring A is said to be strongly regular if for every a e R there exists x ^ R with a -xa 2 (see [5] Symmetric arguments show that (ii) implies (iii) and (iii) implies (ii).…”

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confidence: 99%