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Search: MSC category 16R20 ( Semiprime p.i. rings, rings embeddable in matrices over commutative rings )

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1. CMB 2016 (vol 59 pp. 340)

K╚ępczyk, Marek
A Note on Algebras that are Sums of Two Subalgebras
We study an associative algebra $A$ over an arbitrary field, that is a sum of two subalgebras $B$ and $C$ (i.e. $A=B+C$). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally we generalize this result for semiprime algebras $A$. Consider the class of all semisimple finite dimensional algebras $A=B+C$ for some subalgebras $B$ and $C$ which satisfy given polynomial identities $f=0$ and $g=0$, respectively. We prove that all algebras in this class satisfy a common polynomial identity.

Keywords:rings with polynomial identities, prime rings
Categories:16N40, 16R10, , 16S36, 16W60, 16R20

2. CMB 1998 (vol 41 pp. 81)

Lanski, Charles
The cardinality of the center of a $\PI$ ring
The main result shows that if $R$ is a semiprime ring satisfying a polynomial identity, and if $Z(R)$ is the center of $R$, then $\card R \leq 2^{\card Z(R)}$. Examples show that this bound can be achieved, and that the inequality fails to hold for rings which are not semiprime.

Categories:16R20, 16N60, 16R99, 16U50

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