The maxit and minit of a ring

Abstract

In a recent paper [2] one of the authors has introduced the concept of module type of a ring, for rings with unit. The object of this paper is to generalize this concept to arbitrary rings, without assuming the existence of a unit. This is easily accomplished for rings with one-sided unit, and we shall define the type of such a ring. Theorem 2.5 gives a relation between this type and the module type of [2], and permits the immediate extension of all results in [2] to rings with one-sided unit.

Type Radicals

“…In [9] the module type was used to construct for a general ring an invariant (called the " maxit " of the ring) which coincides with the module type for rings with unit. In the present paper we shall sharpen the definition of [9] in the following way: we shall extend the module type lattice to a lattice ofmaxits by permitting c and k in (c, k) to take on values a> and 0 respectively, in addition to all positive integers. The order in this extended lattice (and hence the lattice operations) is defined as for module types, noting that c ^ oo for all c and k 10 for all k. We now define the maxit m(R) of a ring R as follows:…”