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There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

Abstract: This paper is motivated by Ito''s paper ((Hokkaido) Math. J. 22 (1993) 373), in which he constructed -relative difference sets in . Chen and Li have shown that , the alternating group of degree 5, has a (30,2,29,14)-relative difference set (see Discrete Math. 256 (2002) 301). These are the only examples of nontrivial relative difference sets in nonsolvable groups known to the author. In this paper, we study relative difference sets in and show that there exist exactly two nontrivial relative difference sets in up to equivalence with the aid of a computer. [Copyright &y& Elsevier]

Abstract: A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, “a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring”. Therefore, an interesting natural question of this sort is “whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?” The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. [Copyright &y& Elsevier]

Abstract: In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the “maximal implies prime” variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study families of Artinian or one-dimensional quotients of a polynomial ring R with a special look to level algebras. Let be the scheme parametrizing graded quotients of R with Hilbert function H. Let be any graded surjection of quotients of R with Hilbert function and , respectively. If (respectively ) and A is a “truncation” of B in the sense that (respectively ) for some , then we show there is a close relationship between and concerning e.g. smoothness and dimension at the points and , respectively, provided B is a complete intersection or provided the Castelnuovo–Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to “non-truncated” Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of Cohen–Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of , , whose general elements are Artinian level algebras of type 2. [Copyright &y& Elsevier]

Let (R,m) be a Noetherian local ring of prime characteristic p. In this paper, as an extension of the concept of pn-basis, we introduce the notion of m-adic pn-basis, and we show that R/Rpn has an m-adic pn-basis for every n (n=1,2,…) if and only if R is a regular local ring. For any Noetherian local ring, further we introduce the concept of m-adic free graded algebra and we give a regularity criterion in terms of the higher differential algebra, and others. [Copyright &y& Elsevier]

In this paper, a criterion on the semi-simplicity of the cyclotomic Brauer algebras is given. This result can be considered as a generalization of Wenzl''s theorem on Brauer algebras [Ann. of Math. 128 (1988) 173–193], which was known as Hanlon–Wales conjecture before. [Copyright &y& Elsevier]