The complement operation on modules has not been argued so far as we know. We observe that it plays an important role in software maintenance, as well as join and meet operations. A de nition of a module and operations on them is given in category theoretic terms; we adopt well-known characterization of complement as a left adjoint to join. The de nition is rst given locally and then extended t...

It has been observed by different authors that QTAG-modules behave very much like torsion abelian groups. In this paper, in section 3, we characterize quasi-essential submodules (Theorem 3.9) and further find a characterization for an h-pure submodule to be a direct summand (Theorem 3.11). In section 4, we obtained a necessary and sufficient condition for a submodule to be contained in a minima...

It’s well known that the functional Hilbert space H2 over the unit ball Bd ⊂ C , with kernel function K(z, ω) = 1 1−z1ω1−···−zdωd , admits a natural A(Bd)-module structure. We show the rank of a nonzero submodule M ⊂ H2 is infinity if and only if M is of infinite codimension. Together with Arveson’s dilation theory, our result shows that Hilbert modules stand in stark contrast with Hilbert basi...

ACVF is the theory of non-trivially valued algebraically closed valued f ields. This theory is the model companion of the theory of valued fields. ACVF does not have elimination of imaginaries in the home sort (the valued field sort). Nevertheless, Haskell, Hrushovski, and Macpherson in [2] were able to find a collection of “geometric sorts” in which elimination of imaginaries holds. Let K be a...

Let R be a commutative ring with unity. And let E unitary R-module. This paper introduces the notion of 2-prime submodules as generalized concept ideal, where proper submodule H module F over is said to if , for r and x implies that or . we prove many properties this kind submodules, then only [N ] E, R. Also, non-zero multiplication module, [K: F] [H: every k such K. Furthermore, will study ba...

We present an algorithm to compute the primary decomposition of a submodule N of the free module Z[x1, . . . , xn]. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of N , i.e. the minimal associated primes of the ideal Ann(Z[x1, . . . , xn]/N ) in Z[x1, . . . , xn] and the...

In this note we show that a Noetherian module has a dual module if and only if it satisfies AB5*. A connection between completeness and AB5* is also established. In this note we relate completeness, quasi-completeness, the A B5* condition, and duality. The main result is that a Noetherian R-module has a dual module if and only if it satisfies A B5*. Throughout this note R will denote a commutat...

The notion of the square submodule of a module M over an arbitrary commutative ring R, which is denoted by RM, was introduced by Aghdam and Najafizadeh in [3]. In fact, RM is the R−submodule of M generated by the images of all bilinear maps on M. Furthermore, given a submodule N of an R−module M, we say that M is nil modulo N if μ(M×M) ≤ N for all bilinear maps μ on M. The main question about t...

Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to Kn for some n > 0. When we replace the scalar field K with a commutative ring A, it is no longer true that every A-module with a finite generating set has a basis: not all modules have bases. But when A is a PID, we get something nearly as good as that: (1) Every submodule of An has a basis ...

Submodule construction is the problem of finding a new submodule which, together with a given submodule, provides a behavior that conforms to a given desired global behavior. A new formulation of this problem and its solution in first-order logic is presented, and it is shown how the known solutions to this problem in the context of various communication paradigms and specification formalisms c...