The Equivalence Between Ultrasemiprime and Ultraprime Under Certain Conditions

Abstract

Mathieu proved that any prime algebra which is also C* - algebra is an ultraprime. Brešar shows that C* - algebra is an ultrasemiprime. This paper is give condtions to ultrasempirime algebras to get ultraprime. Mathieu defined ultraprime algebras by using four equivalent conditions. In his definition of ultraprime algebra, he used the ultrapower of algebra. In contrast, in other identical conditions, he once used sequences; in another occasion, he used a particular linear operator. Our proof adds a new condition by using the prime algebra to the ultrasemiprime algebra to get an equivalent condition to ultraprime algebras.

Keywords

• ultraprime algebra
• ultrasemiprime algebra
• ultrapower algebra