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2012•Journal of Algebra and Its ApplicationsRequires access

REGULARITY AND STRONG REGULARITY IN THE CONTEXT OF CERTAIN CLASSES OF RINGS

Michał Ziembowski

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Abstract

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

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What this paper is about

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

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Available abstract

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

Key concepts: Mathematics, Primitive ring, Ring (chemistry), Principal ideal ring, Von Neumann regular ring, Context (archaeology), Noncommutative ring, Combinatorics

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