On group gradings on PI-algebras

We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] \leq exp(W)^K$. A G-grading $W = \bigoplus_{g \in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} \neq 0$ for any $r \geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.Comment: 17 page

The conservative matrix field

We present a new structure called the "conservative matrix field", initially developed to elucidate and provide insight into the methodologies employed by Ap\'ery's in his proof og the irrationality of the Riemann zeta function at 3. This framework is also applicable to other well known mathematical constants, such as e, {\pi}, ln(2), and more, and can be used to study their properties. Moreover, the conservative matrix field exhibits inherent connections to various ideas and techniques in number theory, thereby indicating promising avenues for further applications and investigations