On the structure of a relatively free Grassmann algebra

We investigate the multiplicative and $T$-space structure of the relatively free algebra $F^{(3)}$ with a unity corresponding to the identity $\bigl[[x_1,x_2],x_3\bigr]=0$ over an infinite field of characteristic $p>0$. The highest emphasis is placed on unitary closed $T$-spaces over a field of characteristic $p>2$. We construct a diagram containing all basic $T$-spaces of the algebra $F^{(3)}$, which form infinite chains of the inclusions. One of the main results is the decomposition of quotient $T$-spaces connected with $F^{(3)}$ into a direct sum of simple components. Also, the studied $T$-spaces are commutative subalgebras of $F^{(3)}$; thus, the structure of $F^{(3)}$ and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics of the case $p=2$. In Appendix, we study nonunitary closed $T$-spaces and the case of a field of zero characteristic.