Surender Jain: Nakayama Fuller rings

Abstract: Nakayama(Ann. of Math. 42 (1941)) and Fuller (Pacific Journal of Mathematics 29, 1 (1969)) showed that over an artinian serial ring every module is a direct sum of uniserial quasi-injective modules. In particular, each right ideal of an artinian serial ring is a finite direct sum of quasi-injective right ideals. A ring with the property that each right ideal is a finite direct sum of quasi-injective right ideals will be called a right Nakayama-Fuller ring. For example, commutative, self-injective rings are Nakayama-Fuller rings. In this tallk, various classes of these rings that include local, simple, prime, right non-singular right artinian, and right serial will be discussed. Prime right self-injective right Nakayama-Fuller rings are shown to be simple artinian. Right artinian, right non-singular, right Nakayama-Fuller rings are upper triangular block matrix rings over rings which are either zero rings or division rings. The Nakayama-Fuller ring property is neither left-right symmetric nor Morita invariant.