Some variations of projectivity

We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.

Schanuel's Lemma in P-Poor Modules

Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from a ring with identity, rather than a field. In module theory there is a concept about projective module, i.e. a module over ring R in which it is projective module relative to all modules over ring R. Next, there is the fact that every module over ring R is projective module relative to all semisimple modules over ring R. If P is a module over ring R which it’s projective relative only to all semisimple modules over ring R, then P is called p-poor module. In the discussion of the projective module, there is a lemma associated with the equivalence of two modules K1 and K2 provided that there are two projective modules P1 and P2 such that is isomorphic to . That lemma is known as Schanuel’s lemma in projective modules. Because the p-poor module is a special case of the projective module, then in this paper will be discussed about Schanuel’s lemma in p-poor modules

p-semisimple modules and type submodules

In this paper, we introduce the concept of [Formula: see text]-semisimple modules. We prove that a multiplication reduced module is [Formula: see text]-semisimple if and only if it is a Baer module. We show that a large family of abelian groups are [Formula: see text]-semisimple. Furthermore, we give a topological characterizations of type submodules (ideals) of multiplication reduced modules ([Formula: see text]-semisimple rings). Moreover, we observe that there is a one-to-one correspondence between type ideals of some algebraic structures on one hand and regular closed subsets of some related topological spaces on the other hand. This also characterizes the form of closed ideals in [Formula: see text].