Let (R; m;k) be a local ring (commutative and Noetherian). We will discuss existence and uniqueness of direct-sum decompositions of nitely generated R-modules. One says that R has nite CM type provided there are only nitely many indecomposable maximal Cohen-Macaulay R-modules up to isomorphism. Among complete equicharacteristic hypersurface rings with k algebraically closed of characteristic 6 ...

Let R be a commutative Noetherian ring of dimension d and M cancellative torsion-free seminormal monoid. Then: (1) A type R[d,m,n] P projective A[M]-module rank r≥max {2,d+1}. Then the action E(A[M]⊕P) on (A[M]⊕P) is transitive (2) Assume (R,m,K) regular local containing field k such that either char k=0 or k=p tr-deg K∕

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e...

We prove that there exist a dimension group G whose positive cone is not isomorphic to the dimension monoid DimL of any lattice L. The dimension group G has an order-unit, and can be taken of any cardinality greater than or equal to א2. As to determining the positive cones of dimension groups in the range of the Dim functor, the א2 bound is optimal. This solves negatively the problem, raised by...

In the study of algebraic groups the representative functions related to monoid algebras over fields provide an important tool which also yields the finite dual coalgebra of any algebra over a field. The purpose of this note is to transfer this basic construction to monoid algebras over commutative rings R. As an application we obtain a bialgebra (Hopf algebra) structure on the finite dual of t...

The paper studies the following question: Given a ring R, when does the zero-divisor graph (R) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then (R) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z2 × Z2 × Z2; Z2 × Z4; Z2 × (Z2[x]/(x)); F1 × F2, where F1, F2 are fields. In addition, we determine a...

1 Rings Definition 1.1 (Rings). A ring R is a set with two binary operations + and · called addition and multiplication respectively such that: (1) (R, +) is an Abelian group with identity element 0. (2) (R, ·) is a monoid with identity element 1. (3) Multiplication distributes over addition: a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a for all a, b, c ∈ R. A ring is called commu...