Search: MSC category 16N40
( Nil and nilpotent radicals, sets, ideals, rings )
1. CMB Online first
||A note on algebras that are sums of two subalgebras|
We study an associative algebra $A$ over an arbitrary field,
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
Keywords:rings with polynomial identities, prime rings
Categories:16N40, 16R10, , 16S36, 16W60, 16R20
2. CMB 2004 (vol 47 pp. 343)
||Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras |
We construct new examples of non-nil algebras with any number of
generators, which are direct sums of two
locally nilpotent subalgebras. Like all previously known examples, our examples
are contracted semigroup algebras and the underlying semigroups are unions
of locally nilpotent subsemigroups.
In our constructions we make more
than in the past the close relationship between the considered problem
and combinatorics of words.
Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words
Categories:16N40, 16S15, 20M05, 20M25, 68R15
3. CMB 1998 (vol 41 pp. 79)
||An answer to a question of Kegel on sums of rings |
We construct a ring $R$ which is a sum of two subrings
$A$ and $B$ such that the Levitzki radical of $R$ does not
contain any of the hyperannihilators of $A$ and $B$. This
answers an open question asked by Kegel in 1964.