Generalizations of projectivity and supplements revisited for superfluous ideals

Abstract

We (re)introduce four ideal-related generalizations of classic module-theoretic notions: the ideal-superfluity, projective ideal-covers, the ideal-projectivity, and ideal-supplements. For a superfluous ideal I, the main theorem asserts the equivalence between the conditions: “I-supplements are direct summands in finitely generated projective modules”; “finitely generated I-projective modules are projective”; “projective modules with finitely generated factors modulo I are finitely generated”; “finitely generated flat modules with projective factors modulo I are projective.” Moreover, we provide a property of the ideal I which is sufficient for the equivalence to hold true. The property is expressed in terms of idempotent-lifting in matrix rings.

Keywords:

• Lifting idempotents
• matrix rings
• projective modules
• superfluous submodules
• supplement

2010 MATHEMATICS SUBJECT CLASSIFICATION::

• Primary 16D40
• Secondary 16N40

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by SVV [SVV-2017-260456].