We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups O± m(3) with m ≥ 6 on nonsingular points of their standard modules. These actions together with those studied in [2] are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.

Given a Young diagram λ and the set Hλ of partitions of {1, 2, . . . , |λ|} of shape λ, we analyze a particular S|λ|-module homomorphism QH λ → QHλ ′ to show that QHλ is a submodule of QHλ ′ whenever λ is a hook (n, 1, 1, . . . , 1) with m rows, n ≥ m, or any diagram with two rows.

Let k[V ] be the space of functions from a finite vector space into the algebraically closure of its field of scalars. This paper describes the lattice of subspaces of k[V ] which are invariant under the affine group AGL(V ). The description provides a simple method for finding the submodule generated by any set of functions given as polynomials in the standard coordinates.

Let R be a universally catenary locally equidimensional Noetherian ring. We give a multiplicity based criterion for an arbitrary finitely generated R-module to be integral over a submodule. Our proof is self-contained and implies the previously known numerical criteria for integral dependence of ideals and modules.

In this paper, we investigate the ideal theory in BCK-algebras and we define the notions of primary and P -primary ideals. Then we show that in bounded implicative BCK-algebras, if an ideal has a primary decomposition, then it has a reduced primary decomposition. In the follow, we extend the above concepts to X-modules (Extended BCK-modules) and we introduce the notions of primary and P -primar...

A set of polynomials is called a submodule $\mathbb C [x_1 ,\ldots , x_n ]$ if it translation invariant linear subspace ]$. We present description the submodules [x,y]$ in terms