13 problems

Converse conjecture. The converse of Proposition 3.14 is false: not every semi-inner automorphism of $(_{R}\mathcal{M}^{0})$ is equinumerous.

Existence conjecture. There exists such an $R$ having a semi-inner, non-equinumerous automorphism $\varphi\in\operatorname{Aut}(_{R}\mathcal{M}^{0})$.

For a numerical semigroup $S$, let $g(S)$ denote its genus and, for each $r\in\mathbb{N}_0$, let … where $\widetilde{\Delta}=\Delta\cup\{0\}$ is the numerical semigroup associated…

Let $S$ be a semiring and let $M$ be an $S$-semimodule. A right congruence on $S$ is regular if it is compatible with the right action of $S$, and it is maximal regular if it is ma…

Bound conjecture. There exists a semimodule $\Delta$ minimally generated by $[0,g]$ for a gap $g$ of $\Gamma$, such that

Let ${\mathcal A}$ be the ambient algebraic structure, and let the category of systemic ${\mathcal A}$-modules be the category whose objects are systemic ${\mathcal A}$-modules and…

Il'in's conjecture. Every left (right) $S$-semimodule has an injective hull if and only if $S$ is additively regular. This conjecture characterizes semirings for which the category…

Let $S$ be a semiring, and let $Inj(_{S}\mathcal{M})$ denote the class of injective left $S$-semimodules and $e$-$Inj(_{S}\mathcal{M})$ the class of e-injective left $S$-semimodule…

Let $G=B_1\cup\dotsm\cup B_q$ be a finite partition of a semimodule $(G,\pmb{+})$ over a semiring $(R,+,\cdot)$. For a subset $F\subseteq R$, write $Fb=\{fb:f\in F\}$. Schur–Brauer…

A semiring $S$ is zerosumfree if $a+b=0$ implies $a=b=0$, and it is a CP-semiring if every cyclic right $S$-semimodule is projective. An element $\infty\in S$ is an infinite elemen…

Let $(R,\vee,\circ)$ be a finite simple additively idempotent semiring with an absorbing greatest element, and suppose it possesses a finite idempotent irreducible semimodule witho…

Let $(R,+,\cdot)$ be a finite, simple, additively idempotent semiring, and let $(M,+)$ be a finite idempotent $R$-semimodule. A semimodule is sub-irreducible if it is non-quasitriv…

Perfectness conjecture. In the class of all additively regular semirings, perfect semirings are precisely perfect rings.