‎Let $R$ and $S$ be rings‎, ‎$C= {}_SC_R$ a faithfully SEMIDUALIZING bimodule‎, ‎and $m$ a non-negative integer‎. ‎In this paper‎, ‎we present some homological properties relative to the class of $R$-MODULES of $C$-weak injective dimension at most $m$ and the class of $S$-MODULES of $C$-weak flat dimension at most $m$‎. ‎We introduce and study weak cotorsion MODULES with respect to $m$ and $C$ by using the class of MODULES of $C$-weak flat dimension at most $m$.

All rings throughout this paper are commutative and Noetherian. SEMIDUALIZING MODULES were studied independently by Foxby [4], Golod [5], and Vasconcelos [11]. A finite R-module C is called SEMIDUALIZING if the natural homothety map is an isomorphism and for all . If a SEMIDUALIZING R-module has finite injective dimension, it is called dualizing and is denoted by D. The ring itself is an example of a SEMIDUALIZING R-module. Many researchers, in particular Sather-Wagstaff [12], have studied the SEMIDUALIZING MODULES.      Let M be an R-module. The trace ideal of M, denoted by , is the sum of images of all homomorphisms from M to R. Trace ideals have attracted the attention of many researchers in recent years. In particular, Herzog et al. [6] and Dao et al. [3] studied the trace ideals of canonical MODULES. Also, the trace ideals of SEMIDUALIZING MODULES were studied in [1].        In this paper, we study the trace ideals of tensor product of two arbitrary MODULES. We prove some known facts with a different approach via trace ideals. For example, let C and be two SEMIDUALIZING R-MODULES. We show that is projective if and only if C and  are projective R-MODULES of rank 1. Also, we study the trace ideals of maximal Cohen-Macaulay MODULES over a Gorenstein local ring.

Let $C= {}_SC_R$ be a (faithfully) SEMIDUALIZING bimodule. This paper begins with the introduction of the concepts of $C$-$fp_n$-injective $R$-MODULES and $C$-$fp_n$-flat $S$-MODULES. Subsequently, we investigate various properties associated with classes of MODULES characterized by $C$-$fp_{n}$-injective and $C$-$fp_{n}$-flat dimensions. For instance, we explore Foxby equivalence and the existence of preenvelopes and covers in relation to these classes of MODULES. Finally, we analyze the exchange properties of these classes and the connections between preenvelopes (or precovers) and Foxby equivalence, particularly within the context of almost excellent extensions of rings.

Inspired by a recent work of Buchweitz and Flenner, we show that, for a SEMIDUALIZING bimodule C, C-perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of MODULES which admit minimal resolutions of C-projective MODULES.

Let R be a commutative ring and M be a unitary R- module. We define a semiprime submodule of a module and consider various properties of it. Also we define semi-radical of a submodule of a module and give a number of its properties. We define MODULES which satisfy the semi-radical formula (s.t .s.r.f) and present the existence of such a module.