An $R$-module $M$ has an almost trivial dual if there are no epimorphisms from $M$ to the free $R$-module of countable infinite rank $R^{\left(\omega \right)}$. For every natural number $k>1$, we construct arbitrarily large separable ${\aleph }_k$-free $R$-modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_C$-injective module $G$, the character module $G^{+}$ is $G_C$-flat, then the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$...

A cardinal function $\varphi$ (or a property $𝒫$) is called $l$-invariant if for any Tychonoff spaces $X$ and $Y$ with $C_p\left(X\right)$ and $C_p\left(Y\right)$ linearly homeomorphic we have $\varphi \left(X\right)=\varphi \left(Y\right)$ (or the space $X$ has $𝒫$ ($\equiv X⊢𝒫$) iff $Y⊢𝒫$). We prove that the hereditary Lindelöf number is $l$-invariant as well as that there are models of $ZFC$ in which hereditary separability is $l$-invariant.

The fixed infinitely differentiable function $\rho \left(x\right)$ is such that $\left\{n\rho \right(nx\left)\right\}$ is a regular sequence converging to the Dirac delta function $\delta$. The function ${\delta }_{𝐧}\left(𝐱\right)$, with $𝐱=(x_1,\cdots ,x_m)$ is defined by $${\delta }_{𝐧}\left(𝐱\right)=n_1\rho \left(n_1{x}_1\right)\cdots n_m\rho \left(n_m{x}_m\right).$$ The product $f\circ g$ of two distributions $f$ and $g$ in ${𝒟}_m^{\text{'}}$ is the distribution $h$ defined by $$\underset{n_1\to \infty }{error}\cdots \underset{n_m\to \infty }{error}\langle f_{𝐧}{g}_{𝐧},\phi \rangle =\langle h,\phi \rangle ,$$ provided this neutrix limit exists for all $\phi \left(𝐱\right)={\phi }_1\left(x_1\right)\cdots {\phi }_m\left(x_m\right)$, where $f_{𝐧}=f*{\delta }_{𝐧}$ and $g_{𝐧}=g*{\delta }_{𝐧}$.

An $𝒮$-closed submodule of a module $M$ is a submodule $N$ for which $M/N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $𝒮$-closed submodule $N$ of $M$ is a direct summand of $M$. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$-modules are projective if and only if all right $R$-modules are GCS-modules.