A NOTE ON SEMISTAR OPERATIONS AND LOCALIZING SYSTEMS

Akira Okabe (Japan)

Abstract

In [M. Fontana and J. A. Huckaba, Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 2000, pp. 169-197], the notions of a spectral semistar operation, a stable semistar operation and a quasi-spectral semistar operation were introduced. In this note we introduce three notions of a pseudo-spectral semistar operation, an almost canonically spectral semistar operation, and a canonically spectral semistar operation and investigate their properties and relations among those semistar operations of the above six types. In particular, we shall give some characterizations of an almost canonically spectral semistar operation. We also give a partial answer to a Fontana-Huckaba’s open problem.

Keywords and phrases: semistar operation, localizing system, spectral, quasi-spectral, canonically spectral, Fontana-Huckaba’s open problem.