Links and Blocks

Abstract

Artinian rings.Let A be an Artinian ring. Recall that A is said to beindecomposableif A cannot be split as the direct sum of two non-zero rings. In general we can decompose A as a finite direct sum of indecomposable (and necessarily Artinian) rings. Equivalently, where the ei arepairwise orthogonal primitive central idempotents.(Central idempotents areorthogonalif their product is zero; a central idempotent isprimitiveif it is non-zero and it cannot be written as the sum of two orthogonal non-zero central idempotents.) It’s clear how to get (2) from (1), and to get (1) from (2), take A_i= Ae_i. These decompositions areunique(up to permutation of the indices) — indeed if Ω is any ring summand of A then there is a nonempty subsetJ _Ωof 1,…,t such that

In fact, We call A_i…, A_t theblocks of A.