1. CMB 2016 (vol 59 pp. 340)
 Kȩpczyk, 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 
