Quasiregular element
This article addresses the notion of quasiregularity in the context of ring theory, a branch of modern algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular.
In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring.[1] Intuitively, quasiregularity caputures what it means for an element of a ring to be "bad"; that is, have undesirable properties.[2] Although a "bad element" is necessarily quasiregular, quasiregular elements need not be "bad," in a rather vague sense. The notion of quasiregularity may only be defined for rings with unity. In this article, therefore, all rings are assumed to be unital.
Formal definition
Let R be a ring (with unity) and let r be an element of R. Then r is said to be quasiregular, if 1 - r is a unit in R; that is, invertible under multiplication.[3] The notions of right or left quasiregularity correspond to the situations where 1 - r has a right or left inverse, respectively.[4]
Properties
- If r is nilpotent, then r is quasiregular. An easy computation demonstrates that the inverse of 1 - r is 1 + r + r2 + ... + rn-1, where n is such that rn = 0.[5][6]
- Every element of the Jacobson radical of a (not necessarily commutative) ring is quasiregular.[7] In fact, the Jacobson radical of a ring can be characterized as the unique right ideal of the ring, maximal with respect to the property that every element is right quasiregular.[8][9] However, a right quasiregular element need not necessarily be a member of the Jacobson radical.[10] This justifies the remark in the beginning of the article - "bad elements" are quasiregular, although quasiregular elements are not necessarily "bad." Elements of the Jacobson radical of a ring, are often deemed to be "bad."
- If an element of a ring is nilpotent and central, then it is a member of the ring's Jacobson radical.[11] This is because the principal right ideal generated by that element consists of quasiregular (in fact, nilpotent) elements only.
- If an element, r, of a ring is idempotent, it cannot be a member of the ring's Jacobson radical.[12] This is because idempotent elements cannot be quasiregular. This property, as well as the one above, justify the remark given at the top of the article that the notion of quasiregularity is computationally convenient when working with the Jacobson radical.[13]
Notes
- ^ Isaacs, p. 180
- ^ Isaacs, p. 179
- ^ Isaacs, p. 180
- ^ Isaacs, p. 180
- ^ Isaacs, Problem 12.10, p. 173
- ^ Isaacs, p. 180
- ^ Isaacs, Theorem 13.4(a), p. 180
- ^ Isaacs, Theorem 13.4(b), p. 180
- ^ Isaacs, Corollary 13.7, p. 181
- ^ Isaacs, p. 181
- ^ Isaacs, Corollary 13.5, p. 181
- ^ Isaacs, Corollary 13.6, p. 181
- ^ Isaacs, p. 180
References
- I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.
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