  In this paper we study matrix algebras with a degenerate trace in the
framework of the theory of polynomial identities. The first part is devoted to
the study of the algebra $D_n$ of $n \times n$ diagonal matrices. We prove
that, in case of a degenerate trace, all its trace identities follow by the
commutativity law and by pure trace identities. Moreover we relate the trace
identities of $D_{n+1}$ endowed with a degenerate trace, to those of $D_n$ with
the corresponding trace. This allows us to determine the generators of the
trace T-ideal of $D_3$.
  In the second part we study commutative subalgebras of $M_k(F)$, denoted by
$C_k$ of the type $F + J$ that can be endowed with the so-called strange
traces: $tr(a+j) = \alpha a + \beta j$, for any $a+j \in C_k$, $\alpha$, $\beta
\in F$. Here $J$ is the radical of $C_k$. In case $\beta = 0$ such a trace is
degenerate, and we study the trace identities satisfied by the algebra $C_k$,
for every $k \geq 2$. Moreover we prove that these algebras generate the
so-called minimal varieties of polynomial growth.
  In the last part of the paper, devoted to the study of varieties of
polynomial growth, we completely classify the subvarieties of the varieties of
algebras of almost polynomial growth introduced in an earlier paper of the same
authors.

Rings and Algebras