Abstract: V. V. Shchigolev has proved that over any infinite field $k$ of characteristic $p>2$, in the free associative $k$-algebra on a countable number of generators $\{x_n \mid n \ge 1 \}$, the $T$-space generated by $G=\{ x_n^p \mid n \ge 1 \}$ is finitely based, thus answering a question raised by A. V. Grishin. Shchigolev then conjectured that every infinite subset of $G$ generates a finitely based $T$-space. We prove Shchigolev's conjecture is correct over any field of characteristic $p>2$. We also give an upper bound for the size of a minimal generating set for such a $T$-space.

Abstract: Algebras with polynomial identities generalize commutative and finite-dimensional algebras. This talk will consist of two parts. Part I examines certain Burnside-type conditions on the multiplicative and the adjoint semigroups associated with an associative algebra $A$. A semigroup $S$ is called $n$-collapsing if, for every $s_1,\ldots,s_n$ in $S$, there exist functions $f\neq g$ (depending on $s_1,\ldots, s_n$), such that

$$s_{f(1)}\cdots s_{f(n)} = s_{g(1)}\cdots s_{g(n)}.$$

More specifically, $S$ is called $n$-rewritable if $f$ and $g$ can be taken to be permutations. Semple and Shalev extended Zelmanov's solution of the Restricted Burnside Problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent. In Part I of this talk, we will consider when the multiplicative semigroup of an associative algebra is collapsing; in particular, we prove the following conditions are equivalent, for all unital algebras $A$ over an infinite field: the multiplicative semigroup of $A$ is collapsing, $A$ satisfies a multiplicative semigroup identity, and $A$ satisfies an Engel identity. Furthermore, we will see that, if the multiplicative semigroup of $A$ is rewritable, then $A$ must be commutative.

In Part II of this talk, we will consider algebraic analogues to well-known problems of Philip Hall on verbal and marginal subgroups of a group. Consider the canonical descending and ascending central series of ideals of an associative algebra $A$:

$$A=A^{(1)}\supseteq A^{(2)}\supseteq \cdots \supseteq A^{(n)}\supseteq \cdots\supseteq 0\quad\text {and}$$ $$0=F^{(0)}(A)\subseteq F^{(1)}(A)\subseteq\cdots \subseteq F^{(n)}(A) \subseteq \cdots \subseteq A.$$

Jennings proved that $A^{(n+1)}=0$ precisely when $A=F^{(n)}(A)$. First we will prove that, if $A/F^{(n)}(A)$ is finite-dimensional, then so is $A^{(n+1)}$. This result is an analogue of a group-theoretic result of Baer, which was proved first by Schur in the case when $n=1$. We also will see that the converse holds whenever $A$ is finitely generated. While this is not true for arbitrary algebras $A$, we do show that, if $A^{(n+1)}$ is finite-dimensional, then at least the quotient $A/F^{(3n-1)}(A)$ is finite-dimensional. These two partial converses are analogues of group-theoretic results due to Hall, albeit with a different bound in the second result. Our main technique is to first describe the ideals $A^{(n+1)}$ and $F^{(n)}(A)$ as the verbal and marginal subspaces of $A$ corresponding to a certain polynomial $g_n$ and then apply Stewart’s result in the algebraic analogue of Hall’s First Problem. Moreover, we will consider algebraic analogues to the other Hall’s Problems and will support them positively in some cases.