3 problems

Let $n\in\mathbb{N}$ with $n>1$, and let $\varphi:F\to M_n(G)$ be a ring embedding, where $G$ and $F$ are division rings. Suppose that $a$ and $b$ are positive integers satisfying…

Let $R$ be an arbitrary noetherian PI-ring, and let $(M_n)_{n\in\mathbb{N}}$ be any sequence of pairwise non-isomorphic finitely generated, indecomposable left $R$-modules. Conside…

Let $R$ be a ring. Say that $R$ satisfies the product condition if, for every family $(M_n)_{n\in\mathbb N}$ of pairwise non-isomorphic finitely generated, indecomposable left $R$-…