Quasi-injective modules over valuation domains are classified by means of complete sets of cardinal invariants.

Let $𝔽$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $𝔽$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,...,n_k\le p-1$ such that $V\cong {\oplus }_{i=1}^k{V}_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $S{L}_2\left(ℂ\right)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring...

We investigate the invariant rings of two classes of finite groups $G\le \mathrm{GL}(n,F_q)$ which are generated by a number of generalized transvections with an invariant subspace $H$ over a finite field $F_q$ in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with...

We study the minimal number of elements of maximal order occurring in a zero-sumfree sequence over a finite Abelian p-group. For this purpose, and in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, our method implies that, if we denote by exp(G) the exponent of the finite Abelian p-group G considered, every zero-sumfree sequence S with maximal possible length over...

In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I+J)(I\cap J)=IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family...

Let $R$ be a one-dimensional analytically irreducible ring and let $I$ be an integral ideal of $R$. We study the relation between the irreducibility of the ideal $I$ in $R$ and the irreducibility of the corresponding semigroup ideal $v\left(I\right)$. It turns out that if $v\left(I\right)$ is irreducible, then $I$ is irreducible, but the converse does not hold in general. We collect some known results taken from [5], [4], [3] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition...

We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an (n-1)-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.

We describe the isolated points of an arbitrary topological space $(X,\tau )$. If the $\tau$-specialization pre-order on $X$ has enough maximal elements, then a point $x\in X$ is an isolated point in $(X,\tau )$ if and only if $x$ is both an isolated point in the subspaces of $\tau$-kerneled points of $X$ and in the $\tau$-closure of $\left\{x\right\}$ (a special case of this result is proved in Mehrvarz A.A., Samei K., On commutative Gelfand rings, J. Sci. Islam. Repub. Iran 10 (1999), no. 3, 193–196). This result is applied to an arbitrary subspace of the prime...