The structure of matrix polynomial algebras

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions a coefficient by (elementary) matrix-variables over ground ring in not necessary commuting variables. These subalgebras full rings show up noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining precise lift Hilbert Basis Theorem when is either commutative ring, free skew extension term-ordered monoid. equally address natural but rather delicate question recognising which algebras are Cayley-Hamilton algebras, interesting arising from $\mathrm{Gl}_{n}$-varieties.