We define the notion of " projective " multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra C(T n) of continuous complex-valued functions on an n-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyse...

In this paper, we study the left orthogonal class of max-flat modules which are homological objects related to s-pure exact sequences and module homomorphisms. Namely, a right $A$ is called MF-projective if ${Ext}^{1}_{R}(A,B)=0$ for any $R$-module $B$, strongly ${Ext}^{i}_{R}(A,B)=0$ all $R$-modules $B$ $i\geq 1$. Firstly, give some properties $MF$-projective SMF-projective modules. Then intro...

There is much we still do not know about projective spaces. We describe here how the mod two cohomology of each real projective space is built as an unstable module over the Steenrod algebra A, or equivalently, over K, the algebra of inherently unstable mod two “lower operations” originally introduced by Steenrod. In particular, to produce the cohomology of projective space of each dimension we...

The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex. 0. Introduction This paper proves over a wide class of rings that the Gorenstein projective modules form a precovering class in the module category. Let me explain this statement. There are two terms of mystery, “Gorenstein projective modules” and “precovering...

Let R be a ring and M a right R-module. M is called n-FP-projective if Ext M N = 0 for any right R-module N of FP-injective dimension ≤n, where n is a nonnegative integer or n = . R M is defined as sup n M is n-FP-projective and R M = −1 if Ext M N = 0 for some FP-injective right R-module N. The right -dimension r -dim R of R is defined to be the least nonnegative integer n such that R M ≥ n im...

 Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals o...

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra A $A$ with radical J $J$ will be said to short provided 3 = 0 $J^3 0$ . As case, we show: If a has an indecomposable non-projective Gorenstein-projective module M $M$ , then either is self-injective (so that all modules are Gorenstein-projective) and then, of cou...

It is proved that every commutative ring whose RD-injective modules are Σ-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian (respectively simple) module is RD-injective, is given. These results can be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively th...