Subdirect sum decompositions of endomorphism rings

Abstract

In [4] Levy defined irredundant subdirect sums of rings. This provided a unique decomposition of a left quotient semisimple ring as a subdirect sum of left quotient simple rings. He also showed that irredundant subdirect sums of prime rings possess some rather strong uniqueness properties. In [2] essential subdirect sums were defined and Levy's decomposition theorem was obtained for essential subdirect sums. Furthermore in [3] it was shown that a ring R has a maximal left quotient ring which is a direct product of full linear rings if and only if R is an essential subdirect sum of rings whose maximal left quotient ring is full linear. In §1 of this paper we show that for subdirect sums of prime rings, essential and irredundant subdirect sums are identical. We also give some uniqueness properties of essential subdirect sums which are even stronger than those for irredundant subdirect sums. In §2 we show that if M is a torsionless 12-module and 22 is an essential subdirect sum of prime rings, then the endomorphism ring of M is also an essential subdirect sum of prime rings. We get this theorem as a consequence of a more general result about Morita contexts. The proof of the more general result is notationally more efficient. 1* Essential subdirect sums of prime rings* In what follows R will be an associative ring which is not assumed to have an identity. RM will express that M is a left JS-module. A submodule K of M is an essential submodule if every nonzero submodule of M has nonzero intersection with K. M will then be an essential extension