Topic
Irreducible ring
About: Irreducible ring is a research topic. Over the lifetime, 13 publications have been published within this topic receiving 126 citations.
Papers
TL;DR: In this paper, a general theory of simple rings, both associative and non-associative, is presented, which is essentially as complete as that of the classical case of rings that satisfy the descending chain condition for right ideals.
Abstract: It is the purpose of this paper to lay the foundations of a general theory of simple rings, both associative and non-associative. In part I we obtain the structure of simple associative rings that either contain minimal right ideals or contain maximal right ideals. In either case we obtain a realization of our ring 2f as a certain type of ring of linear transformations in a vector space over a division ring. If 2{ has a minimal right ideal this realization is essentially unique and we can prove a converse theorem that rings of linear transformations having certain properties are simple and contain minimal right ideals. Thus the theory here is essentially as complete as that of the classical case of rings that satisfy the descending chain condition for right ideals. Our structure theory hinges on a general theorem (Theorem 6) on the structure of irreducible rings of endomorphisms of commutative groups. This theorem may be regarded as an extension of Burnside's theorem on irreducible algebras of matrices. The classical theory of simple rings with descending chain condition is a simple consequence of our results and we believe that the present treatment is more transparent than the methods of proof previously given(1) . In studying an arbitrary non-associative ring 9t one is led to consider the associative ring 92 generated by the left and the right multiplications x->ax and x-xcxa acting in 2. If 21 is simple, 91 is an irreducible ring of endomorphisms. In part II we describe the structure of $1 in terms of the multiplication centralizer ( of 2[ defined to be the totality of endomorphisms 'y in 2 such that (xy)y=(xy)y=x(yy). We define the concepts of center, central algebra and extension of the underlying field of an algebra. In Part III we investigate a special type of simple associative algebra that may be regarded as a generalization of the concept of the complete algebra of linear transformations in a vector space over a field, or equivalently, of the concept of the complete matrix algebra. There are many points of contact between the discussion here (and in other parts of the paper) and recent work in the theory of rings of transformations in Banach spaces and in vector spaces over the fields of reail and complex numbers(2).
64 citations
TL;DR: In this paper, it was shown that a primitive ring can be considered to be left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms.
Abstract: In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.
31 citations
1 Apr 1957
TL;DR: In this article, it was shown that every commutative subdirectly irreducible ring is a subdirect sum of subdirectively irreduceible rings, some of which are fields, the others being bound to their maximal nilideals.
Abstract: Introduction. A subdirectly irreducible ring is one in which the intersection of all the nonzero ideals is a nonzero ideal. Such rings are important not only because every ring is isomorphic to a subdirect sum of subdirectly irreducible rings, but also because the theory of rings without chain conditions uses the concept heavily. Our major knowledge of such rings is contained in [i 1, where Professor McCoy showed that every commutative subdirectly irreducible ring is one of three kinds. We shall classify them as Type (a). Fields. Type (p). Every element is a divisor of zero. Type (y). There exist both nondivisors of zero and nilpotent elements. We shall restrict ourselves to the commutative case and henceforth not repeat its assumption. We shall employ the following notation: A =the commutative subdirectly irreducible ring. D = the set of all divisors of zero of A. J = the Jacobson radical of A. N=the maximal nilideal of A. N*= the set of elements that annihilate N. D*=the set of elements that annihilate D. Q= the unique minimal ideal of A that is contained in every nonzero ideal of A. As in [2] we shall say that A is bound to its maximal nilideal if N*
18 citations
TL;DR: In this paper, a finite non-commutative sub-directly irreducible ring R with heart is shown to be isomorphic with GF(2) + GF (2) (GF(2 is the two element Galois Field) under homomorphisms.
Abstract: Every ring is isomorphic to a subdirect sum of subdirectly irreducible rings Unfortunately, however, as is shown, the property of being subdirectly irreducible is not preserved under homomorphisms An example is given of a finite non-commutative subdirectly irreducible ring R with heart (= the intersection of all non-zero ideals) H , such that R / E is isomorphic with GF(2) + GF(2) (GF(2) is the two element Galois Field) Some additional properties of the ring R are listed and contrasts are made with results for commutative subdirectly irreducible rings; for example, the zero divisors of R do not form an ideal
3 citations
TL;DR: In this paper, it was shown that a commutative sub-directly irreducible ring with minimal ideal M 2 = 0 can be constructed without a unity element.
Abstract: Let R be a commutative subdirectly irreducible ring, with minimal ideal M . It is shown that either R is a field, or M 2 = 0. A construction is given which yields commutative sub-directly irreducible rings possessing nonzero-divisors, and nonzero nilpotent elements either with a unity element, or without. Such a ring without unity has been constructed by Divinsky. The same technique enables the construction of subdirectly irreducible rings with mixed additive groups.
3 citations
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Performance Metrics
| Year | Papers |
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| 2019 | 1 |
| 2016 | 1 |
| 2011 | 1 |
| 1986 | 1 |
| 1984 | 1 |
| 1979 | 1 |