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In this paper we study matrix algebras with a degenerate trace in theframework of the theory of polynomial identities. The first part is devoted tothe study of the algebra $D_n$ of $n \times n$ diagonal matrices. We provethat, in case of a degenerate trace, all its trace identities follow by thecommutativity law and by pure trace identities. Moreover we relate the traceidentities of $D_{n+1}$ endowed with a degenerate trace, to those of $D_n$ withthe corresponding trace. This allows us to determine the generators of thetrace 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 strangetraces: $tr(a+j) = \alpha a + \beta j$, 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 isdegenerate, and we study the trace identities satisfied by the algebra $C_k$,for every $k \geq 2$. Moreover we prove that these algebras generate theso-called minimal varieties of polynomial growth.  In the last part of the paper, devoted to the study of varieties ofpolynomial growth, we completely classify the subvarieties of the varieties ofalgebras of almost polynomial growth introduced in an earlier paper of the sameauthors.

Matrix algebras with degenerate traces and trace identities