π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

Abstract

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. For a given semiprime algebra A, we discuss the π-complementation of the unitisation algebra A ¹ of A. Moreover, if in addition the multiplication algebra ℳ(A) of A is also semiprime, we study the π-complementation in the algebras ℳ(A) and ℳ^♯(A) (the multiplication ideal of A). In associative setting, we prove that A is π-complemented if and only if ℳ^♯(A) is π-complemented, and that A ¹ π-complemented if and only if ℳ(A) is π-complemented.

Key Words:

• Central closure
• Complemented algebra
• Extended centroid
• Multiplication algebra
• Semiprime algebra

2010 Mathematics Subject Classification:

• Primary 17A01
• Secondary 17A60

ACKNOWLEDGMENT

Partially supported by Junta de Andalucía grant FQM 0290 and Project MTM2009-12067.

Notes

Communicated by M. Bresar.