Maximal left ideals and idealizers in matrix rings

Abstract

In this chapter we supply all the necessary definitions as well as the required results needed in this work. All the notation and terminology will also be explained carefully. §1 DEFINITIONS AND NOTATION R will always denote a ring with identity and Mn(R) will denote the ring of nxn matrices over R. As usual the ring of integers, the ring of integers modulo n and the field of rational numbers will be denoted by z, Zn and Q respectively. ng of polynomials in the indeterminate x. The constant term of any polynomial fER[x] will be denoted by const(f). Ideal (or module) will always mean left ideal (or module). In order to simplify notation we shall adopt the convention M,N, M/N, etc. in stead of RM' RN, RM/N, etc., for left Rmodules. It will however always be evident from the context, to which ring R we are referring. Max(R) will denote the collection of all maximal left ideals of R. Mand N will be generic symbols for maximal left ideals. Normally mappings will be written on the left except in the cases of Proposition 1.12 and 1.15. R will be considered as a subring of Mn(R) via the natural embedding r + diag(r, ••• ,r).