Let S = K[x 1 , ..., x n ] or S = K[[x 1 , ..., x n ]] be either a polynomial or a formal power series ring in a finite number of variables over a field K of characteristic p > 0 with [K : K p ] < ∞. Let R be the hypersurface S/f S where f is a nonzero nonunit element of S. If e is a positive integer, F e * (R) denotes the R-algebra structure induced on R via the e-times iterated Frobenius map ( r → r p e ). We describe a matrix factorizations of f whose cokernel is isomorphic to F e * (R) as R-module. The presentation of F e * (R) as the cokernel of a matrix factorization of f enables us to find a characterization from which we can decide when the ring S [[u, v]]/(f + uv) has finite F-representation type (FFRT) where S = K[[x 1 , ..., x n ]]. This allows us to create a class of rings that have finite F-representation type but not finite CM type. For S = K[[x 1 , ..., x n ]], we use this presentation to show that the ring S[[y]]/(y p d + f ) has finite F-representation type for any f in S. Furthermore, we prove that S/I has finite F-representation type when I is a monomial ideal in eitherFinally, this presentation enables us to compute the F-signature of the rings S [[u, v]]/(f + uv) and S[[z]]/(f + z 2 ) where S = K[[x 1 , ..., x n ]] and f is a monomial in the ring S. When R is a Noetherian ring of prime characteristic that has FFRT, we prove that R[x 1 , ..., x n ] and R[[x 1 , ..., x n ]] have FFRT. We prove also that over local ring of prime characteristic a module has FFRT if and only it has FFRT by a FFRT system. This enables us to show that if M is a finitely generated module over Noetherian ring R of prime characteristic p, then the set of all prime ideals Q such that M Q has FFRT over R Q is an open set in the Zariski topology on Spec(R).when I was studying instead of playing with them. Thanks for putting up with me.A special thanks goes to Prof. Salah-Eddine Kabbaj, Prof. Abdeslam Mimouni and Prof. Jawad Abuihlail for the courses in commutative algebra I took with them at King Fahd University of Petroleum and Minerals of Saudi Arabia. Thanks for their encouragement for me to pursue my PhD. I would like to also thank Prof. Craig Huneke for the other courses in commutative algebra that I took with him at the University of Kansas while I was studying my master degree.I am very grateful to King Khalid University of Saudi Arabia for funding my studies abroad. Without their financial support, this would not have been accomplished.Last but not least, I am so grateful for my best friend Nabil Alhakamy for his constant support and encouragement throughout my journey. I would always remember his supportive friendship. I must also thank my classmate Mehmet Yesil for being a nice friend. I would like to thank my friends in Sheffield for the treasuring moments we shared together. I find myself lucky of having them during my stay in Sheffield.5 On the FFRT over hypersurfaces 5.1 The presentation of F e * (S/f S) as a cokernel of a Matrix Factorization of f .

A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then [Formula: see text] or [Formula: see text] (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of [Formula: see text], called the class of quasi-[Formula: see text]-ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of [Formula: see text] is a maximal or a minimal ideal of [Formula: see text]. Many examples are given to illustrate the obtained results.

This paper studies properties of certain hypersurfaces in prime characteristic: we give a sufficient and necessary conditions for some classes of such hypersurfaces to have Finite Frepresentation Type (FFRT) and we compute the F -signatures of these hypersurfaces. The main method used in this paper is based on finding explicit matrix factorizations. Recall that a non-zero finitely generated module M over a local ring R is Cohen-Macaulay if depth R M = dim M and R is a Cohen-Macaulay ring if R itself is a Cohen-Macaulay module. However, if depth R M = dim R, M is called maximal Cohen-Macaulay module (or MCM module). When M is a finitely generated module over non-local ring R, M is Cohen-Macaualy if M m is a Cohen-Macaulay module for all maximal ideals m ∈ Supp M.Later in the paper we will look at the modules F e * (R) when R is a Cohen-Macaulay ring. Note that a regular sequence on an R-module M is also a regular sequence on F e * M, and in particular, if R is Cohen-Macaulay, F e * (R) are MCM modules for all e ≥ 0. Definition 2.1. A finitely generated R-module M is said to have finite F-representation type (henceforth abbreviated FFRT) by finitely generated R-modules M 1 , . . . , M s if for every positive integer e, the R-module F e * (M) can be written as a finite direct sum in which each direct summand is isomorphic to some module in {M 1 , . . . , M s } , that is, there exist non-negative integers t (e,1) , . . . , t (e,s) such that F e * (M) = s j=1 M ⊕t (e,j)

In this article, we are interested in uniformly p r pr -ideals with order ≤ 2 \le 2 (which we call 2 r 2r -ideals) introduced by Rabia Üregen in [On uniformly pr-ideals in commutative rings, Turkish J. Math. 43 (2019), no. 4, 18781886]. Several characterizations and properties of these ideals are given. Moreover, the comparison between the (nonzero) 2 r 2r -ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero-divisors. Namely, among other things, we compare the class of (nonzero) 2 r 2r -ideals with the class of (minimal) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of 2 r 2r -ideal in polynomial rings allows us to give a new characterization of the rings satisfying the famous A A -property.