A note on multiplicative primeness of prime degenerate Jordan algebras
An algebra \(A\) is called prime if, given two nonzero ideals \(U\) and \(V\) of \(A\), always \(UV\not=0\). It is well-known that the multiplication algebra \(M(A)\) of a prime algebra need not be a prime algebra. An algebra \(A\) is called multiplicatively prime (m. p.) if both \(A\) and \(M(A)\) are prime algebras. The authors show the primeness of the multiplication algebra of the two algebras introduced by \textit{V. G. Skosyrskii} [Algebra Logic 33, No. 3, 169--179 (1994); translation from Algebra Logika 33, No. 3, 301--316 (1994; Zbl 0826.17037)]. It is shown that the Jordan algebra \(J(G,\delta)\) obtained from the Grassmann algebra by replacing its associate product with a new one defined using odd derivations of \(G\) is a multiplicatively prime degenerate Jordan algebra, and that the Jordan algebra \(J(G,D)\) obtained from the Grassmann algebra by replacing its associate product with a new one defined using even derivations of \(G\) is a multiplicatively prime degenerate Jordan algebra as well.
scientific article; zbMATH DE number 5824084