Marubayashi-Krull orders and strongly graded rings

Abstract

1. INTRODUCTION The aim of this note is to give sufficient conditions for a strongly Z-graded ring to be a Marubayashi-Krull order (cf. Theorem 3.10). Concrete examples of strongly E-graded rings are skew group rings over Z and generalized Rees rings. We also prove that under the same conditions the positive part of a strongly Z-graded ring is a Marubayashi-Krull order (cf. Theorem 4.1). Finally, we examine when these conditions may be weakened. 2. PRELIMINARIES A ring R is said to be a strongly B-graded ring if there exists a family of additive subgroups R, (n E Z) of R such that R = OnpLRn and wm =Rn+m for all n, m E Z. The elements of h(R) = lJneh R, are called the homogeneous elements of R. If x E R,, then x is said to be homogeneous of degree n. If x = C x, (x, E R,), then the elements x, are called the homogeneous components of x. Let I be an ideal of R. Denote by Ig the ideal generated by the homogeneous elements contained in I. I is said to be graded if I = Ig. In this case Z= RI, =Z,R, where I, = Zn R,. Details about strongly graded rings may be found in [lo]. If I is an ideal of R, then C(I) denotes the set of elements regular modulo I. If C(I) is a left and right Ore set in R, then Rcc,, denotes the left and right localization of R towards this Ore set C(1). If R is a prime Goldie ring, then Q(R), the classical ring of quotients of R, is a simple Artinian ring. The Asano overring of R is defined as S(R) = {x E Q(R) 1 Ix c R for some nonzero ideal I of R}. A prime Goldie ring R is said to be a Marubayashi-Krull order (or for short an M-Krull order) if: