A left module $M$ over an arbitrary ring is called an $\mathrm{ℛ𝒟}$-module (or an $\mathrm{ℛ𝒮}$-module) if every submodule $N$ of $M$ with $\mathrm{Rad}\left(M\right)\subseteq N$ is a direct summand of (a supplement in, respectively) $M$. In this paper, we investigate the various properties of $\mathrm{ℛ𝒟}$-modules and $\mathrm{ℛ𝒮}$-modules. We prove that $M$ is an $\mathrm{ℛ𝒟}$-module if and only if $M=\mathrm{Rad}\left(M\right)\oplus X$, where $X$ is semisimple. We show that a finitely generated $\mathrm{ℛ𝒮}$-module is semisimple. This gives us the characterization of semisimple rings in terms of $\mathrm{ℛ𝒮}$-modules. We completely determine the structure of these...

Suppose $p$ is a prime number and $R$ is a commutative ring with unity of characteristic 0 in which $p$ is not a unit. Assume that $G$ and $H$ are $p$-primary abelian groups such that the respective group algebras $RG$ and $RH$ are $R$-isomorphic. Under certain restrictions on the ideal structure of $R$, it is shown that $G$ and $H$ are isomorphic.

In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\cdots ,g_k$ with order $n_1,\cdots ,n_k\in \{1,2,\cdots \}\cup \left\{\infty \right\}$, then $${\beta }_1^{\left(2\right)}\left(G\right)\le k-1-\sum _{i=1}^k\frac{1}{n_i},$$ where ${\beta }_1^{\left(2\right)}\left(G\right)$ denotes the first ${\ell }^2$-Betti number of $G$. We also show that any $k$-generated group with ${\beta }_1^{\left(2\right)}\left(G\right)\ge k-1-\epsilon$ must have girth greater than or equal $1/\epsilon$.

We show in an additive inverse regular semiring $(S,+,·)$ with $E^{•}\left(S\right)$ as the set of all multiplicative idempotents and $E^{+}\left(S\right)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e,f\in E^{•}\left(S\right)$, $ef\in E^{+}\left(S\right)$ implies $fe\in E^{+}\left(S\right)$. (ii) $(S,·)$ is orthodox. (iii) $(S,·)$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.