Closed Prime Ideals in Algebras with Semiprime Multiplication Algebra

Abstract

Let A be an algebra whose multiplication algebra M(A) is semiprime. We prove that, except in an exceptional case, the proper closed prime ideals of A are the maximal closed ideals of A, for the closure operations π and ϵ. In fact, these sets agree for both closures. The same can be said in M(A) for the closure operations π and ϵ′. Moreover, we establish the relationships between the proper closed prime ideals of A and the ones of the algebras M(A),U and A/U, for a given ideal U of A.

Key Words:

• Closure operations
• Multiplication algebra
• Prime ideals
• Semiprime algebras

2000 Mathematics Subject Classification:

• Primary 17A60
• Secondary 17A65

ACKNOWLEDGMENTS

This research was partially supported by the Spanish MEC Project MTM2005-02159 and the Junta de Andalucía Grant FQM290.

Notes

Communicated by M. Ferrero.