The maximal regular ideal of a ring

Brown, Bailey; McCoy, Neal H.

doi:10.1090/s0002-9939-1950-0034757-7

Cited by 73 publications

(42 citation statements)

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“…We consider next some connections between condition Cr and the maximal regular ideal [5] of A. We use the following lemma.…”

Section: Theoremmentioning

confidence: 99%

“…Since A is a regular ring, so is M* -7= A. Furthermore, the maximal regular ideal of M*^A -M is zero [5]. It follows then from [5,Theorem 6] that the ring M* is bound to its radical in the sense of M. Hall [lO].…”

Section: Theoremmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…For example, the lattice of all normal subloops of a loop satisfies the conditions of the first statement of Lemma 1. Moreover, each element of this lattice is a meet of meet-irreducible elements (5). Thus the lattice of all normal subloops of a loop G (consisting of more than one element) is complemented if and only if G is the sum of its minimal normal subloops.…”

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confidence: 99%

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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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“…We consider next some connections between condition Cr and the maximal regular ideal [5] of A. We use the following lemma.…”

Section: Theoremmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Ideal lattices and the structure of rings

Blair¹

1953

Trans. Amer. Math. Soc.

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“…It is shown in Section 3 below that abelian VNL rings are precisely those exchange rings R in which, for any idempotent e, one of the two corner rings eRe and (1 − e)R(1 − e) is regular. Let M(R) denote the maximal regular ideal of a ring R as defined by Brown and McCoy in [1]. In ( [3], Lemma 2.7) Chen and Tong showed that if R is an abelian VNL ring, then R/M(R) is a local ring.…”

Section: Introductionmentioning

confidence: 99%

Some Characterizations of VNL Rings

Grover

Khurana

2009

Communications in Algebra

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A ring R is said to be VNL if for any a ∈ R, either a or 1−a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without infinite set of orthogonal idempotents; and also the VNL rings having primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a 1 , a 2 ) ∈ R 2 , one of the a i is regular in R. Formal triangular matrix rings that are VNL, are also characterized. As a corollary it is shown that an upper triangular matrix ring T n (R) is VNL if and only if n = 2 or 3 and R is a division ring.

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The radical property of rings such that every homomorphic image has no nonzero left annihilators

Szász¹

1971

Mathematische Nachrichten

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The fundamental notions, used in this paper, can be found in N. DI-VINSKP [12], N. JACOBSON [17] and N. H. McCoy [is]. All rings, considered here, will be associative. For a radical property R, a ring A is said to be strongly R-semisimple, if any homomorphic image of). Furthermore, a ring A is called strongly idempotent (or antisimple) if any ideal of A is idempotent (or if A cannot be homomorphically mapped onto a subdirectly irreducible ring having idempotent heart, respectively) (cf. V. A. ANDRUNAKIEVITCH [2]). Instance for a strongly idempotent ring is any biregular ring (cf. V. A. ANDRUNA-KIEVITCH [l], in which any principal twosided ideal has twosided unity element, and also any VON KEUMANX regular ring A, in which a E a A a for any element a € A holds. Tho principal left ideal (or twosided ideal), generated by a, of A, will be denoted by (a)( (or by ( a ) , respectively). The ideal a A + A u A for any element a of the ring A 'will be denoted by

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The maximal regular ideal of a ring

Cited by 73 publications

References 3 publications

Ideal lattices and the structure of rings

Ideal lattices and the structure of rings

Some Characterizations of VNL Rings

The radical property of rings such that every homomorphic image has no nonzero left annihilators

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