A local ring is an associative ring with unique maximal ideal. We associate with each Artinian local ring with singleton basis four invariants (positive integers) p,n,s,t. The purpose of this article is to describe the structure of such rings and classify them (up to isomorphism) with the same invariants. Every local ring with singleton basis can be constructed over its coefficient subring by a certain polynomial called the associated polynomial. These polynomials play significant role in the enumeration.
Abstract.A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that A = A/J(A) is a separable field extension of R = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R 0 which is a chain ring such thatThe structure of A in terms of a skew polynomial ring over R 0 is determined. Let R be a commutative chain ring, and A be a local ring that is a faithful R-algebra. Then J(R) = R ∩ J(A). Let A = A/J(A) be a separable, algebraic field extension of R, and let A be either a locally finite Ralgebra or an artinian duo ring. As proved in [1], A has a commutative local R-subalgebra R 0 such thatThis subalgebra R 0 is also called a coefficient subring of A; such a subring is a commutative chain ring, and is a faithful R-algebra. The group of R-automorphisms of R 0 is investigated in Section 2. Wirt [8] introduced the concept of a distinguished basis of a bimodule over a Galois ring. In Section 3 an analogous concept for bimodules over R 0 is investigated.The main purpose of this paper is to prove a representation theorem for A, in case A is a chain ring, in terms of an appropriate homomorphic image of a skew polynomial ring over its coefficient subring. Sections 4 and 5 are devoted to proving the main theorem (Theorem 5.5). By Cohen [5], any
ABSTRACT. According to general terminology, a ring R is completely primary if its set of zero divisors fo.ms an dcal. Let R bc a finite completely primary ring. It is easy to establish that J is the unique maximal ideal of R and R has a coefficient subring S (i.e. R/J isomorphic to S/pS) which is a Galois ring. In this paper we give the construction of finite completely primary rings in which the product of any two zero divisors is in S and determine their enumeration. We also show that finite rings in which the product of any two zero divisors is a power of a fixed prime p are completely primary tings with either J"=0 or their coefficient subring is n with n=2 or 3. A special case of these rings is the class of finite rings, studied in [2], in which the product of any two zero divisors is zero.
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