Characterizations of rings and modules by means of lattices.

PhDIn this thesis we study the relationship between the lattice of submodules and the algebraic structure of a module. The key remark in our study will be the fact that the homomorphisms between two independent submadules of a module can be 'represented' by elements of its lattice of submoduleso Exploiting this fact we show that the endomorphism ring of a module which is the direct sum of more than three isomorphic submodules is determined up to isomorphism by its lattice of submodules. Lattice isomorphisms arise naturally in two ways, viz., through category equivalences and semi-linear isomorphisms. Any lattice isomorphism between a free module of infinite rank and a module containing at least one free submodule is shown to be induced by a category equivalence. This result is used to give new characterizations of Morita equivalence, If certain mild conditions are satisfied a lattice isomorphism between a free module of rank >3 and a faithful module is shown to give rise to a semi-linear isomorphism between the modules* If both nodules are free of rank n>3 then the question of whether there is a semi-linear isomorphism between them is equivalent to asking when an isomorphism. of matrix rings Rn Cý!! Sn implies a ring isomorphism R2ý S. -3- Wo study rings R with this property for any n and any ring S. The following are shown to be of this type (1) commutative rings (2) p-trivial rings (3) matrix rings over strongly regular rings left self-injective rings. Applying these results we give new examples of regular rings which uniquely co-ordinatize a complemented modular lattice of otder In particular we show such a co-ordinatization is always unique to within injective hull

Regular and generalized regular rings

During his study of continuous geometries, J. von Neumann found that any complemented modular lattice satisfying certain mild conditions arises from a specific type of ring, which he termed a regular ring. In the first place, these rings turn out to be a generalization of artinian semisimple rings, and subsequent studies have shown that the concepts of regularity and some of its weaker and stronger forms are characterized by specific ideal structures which give these rings an important place in the general theory. In the first three chapters of this thesis (and in section one of chapter four), many of the known results on regularity and some of its generalizations are collected and arranged. The results are not always presented in chronological order, but are so organised as to show some development of and relationships amongst the various properties of regular and generalized regular rings. Some results which are well-known or less relevant to this development have been stated without proof. Occasionally, proofs have been supplied for results which seem only to be stated in the literature, and at times the original proofs have been modified in the interests of directness or increased generality. In all cases, the proofs of known results carry a reference in brackets, ( ). In the final two sections of the thesis, the author examines some of the properties of weakly regular rings, and defines generalizations of these rings parallel to some of those which have previously been defined for regularity. These new types of weak regularity are found to have several properties analogous to those known for the corresponding types of regularity, and in particular the ws-regular ring (defined on page 51) is characterized by an ideal structure of the same type as those found for strongly regular, regular, and weakly regular rings (as detailed on pages 53 and 54). Finally, on page 56 the author defines weak π-regularity analogously to the definition of π-regularity, and the diagram on page 59 shows how ws-regular and weakly π-regular rings fit into a pattern formed by the above mentioned known types of regularities

Diagonal F-splitting and Symbolic Powers of Ideals

Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$ essentially of finite type over an $F$-finite field. We show that $J^{s+t} \subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})$ for all $s, t, \epsilon > 0$ for which the formula makes sense. We use this to show a number of novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that $P^{(2hn)} \subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.Comment: Many small changes. Notably, I added missing Noetherianity and reducedness assumptions to section 2 and corrected an error in lemma 2.2. Upon reflection, the assumptions on $A$ and $B$ in prop 5.3 were just slightly more general than assuming $A$ and $B$ were fields, so I went ahead and said $A$ and $B$ should be field

FF-pure homomorphisms, strong FF-regularity, and FF-injectivity

We discuss Matijevic-Roberts type theorem on strong $F$-regularity, $F$-purity, and Cohen-Macaulay $F$-injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of $F$-purity of homomorphisms using Radu-Andre homomorphisms, and prove basic properties of it. We also discuss a strong version of strong $F$-regularity (very strong $F$-regularity), and compare these two versions of strong $F$-regularity. As a result, strong $F$-regularity and very strong $F$-regularity agree for local rings, $F$-finite rings, and essentially finite-type algebras over an excellent local rings. We prove the $F$-pure base change of strong $F$-regularity.Comment: 37 pages, updated the bibliography, and modified some error