Extended centroid and central closure of the multiplication algebra

Totally multiplicatively prime algebras

“…We recall that the algebra A is said to be prime if, for ideals U and V of A, U V = 0 implies either U = 0 or V = 0. Following [2], the algebra A is said to be multiplicatively prime whenever both A and M (A) are prime algebras.…”

“…The algebra A is said to be multiplicatively prime whenever both A and M (A) are prime algebras. A characterization of the multiplicative primeness in terms of operators was given in [2], which will be the starting point of this work. Following an already standard process used in [4,5,12], one can consider the corresponding normed strengthening of the quoted algebraic characterization, in this way the totally multiplicatively prime (TMP) algebras appear.…”

A Note on Multiplicative Primeness of Prime Degenerate Jordan Algebras

“…Now, by Proposition 4, we can assert that F (U ) = 0 for every nonzero ideal U of J(G, ∂). Finally, by [11,Proposition 1], we can conclude that J(G, ∂) is m. p.…”

“…x m ∈ U , it follows that F (U ) = 0. Finally, by[11, Proposition 1], we can conclude that J(G, D) is m. p.…”

Associative and Lie algebras of quotients

“…The first examples of dense extensions were given in the context of multiplicatively semiprime algebras. Recall that an algebra A (associative or not) is said to be multiplicatively semiprime if A and the multiplication algebra M (A) (generated by the right and left multiplication operators of A together with the identity) are both semiprime (see [6], [7], [3], [4] among others). Roughly, an extension A ⊆ B of algebras is dense if every non-zero element in M (B) remains non-zero when restricted to A. Cabrera proved in [5] that every essential ideal of a multiplicatively semiprime algebra is dense.…”