This paper presents an introduction to the theory of ideals in a ring with emphasis on ideals in a commutative ring with identity.

The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite global dimension. Very little is known about singularities of noncommutative rings, particularly those which are not finite over their centre. In this thesis, we are able to give a precise description of the singularities of a few families of examples. In many examples, we lay the foundations of noncommutative singularity theory by giving a precise description of the singularities of the fundamental examples of noncommutative surfaces. We draw comparisons with the fundamental examples of commutative surface singularities, called Kleinian singularities, which arise from the action of a finite subgroup of SL(2; k) acting on a polynomial ring. The main tool we use to study the singularities of noncommutative surfaces is the singularity category, first introduced by Buchweitz in [Buc86]. This takes a (possibly noncommutative) ring R and produces a triangulated category Dsg(R) which provides a measure of "how singular" R is. Roughly speaking, the size of this category reflects how bad the singularity is; in particular, Dsg(R) is trivial if and only if R has finite global dimension. In [CBH98], Crawley-Boevey-Holland introduced a family of noncommutative rings which can be thought of as deformations of the coordinate ring of a Kleinian singularity. We give a precise description of the singularity categories of these deformations, and show that their singularities can be thought of as unions of (commutative) Kleinian singularities. In particular, our results show that deforming a singularity in this setting makes it no worse. Another family of noncommutative surfaces were introduced by Rogalski-Sierra-Stafford in [RSS15b]. The authors showed that these rings share a number of ring-theoretic properties with deformations of type A Kleinian singularities. We apply our techniques to show that the "least singular" example has an A1 singularity, and conjecture that other examples exhibit similar behaviour. In [CKWZ16a], Chan-Kirkman-Walton-Zhang gave a definition for a quantum version of Kleinian singularities. These require the data of a two-dimensional AS regular algebra A and a finite group G acting on A with trivial homological determinant. We extend a number of results in [CBH98] to the setting of quantum Kleinian singularities. More precisely, we show that one can construct deformations of the skew group rings A#G and the invariant rings AG, and then determine some of their ring-theoretic properties. These results allow us to give a precise description of the singularity categories of quantum Kleinian singularities, which often have very different behaviour to their non-quantum analogues.

Factorization theory is concerned with the decomposition of mathematical objects. Such an object could be a polynomial, a number in the set of integers, or more generally an element in a ring. A classic example of a ring is the set of integers. If we take any two integers, for example 2 and 3, we know that $2 \cdot 3=3\cdot 2$, which shows that multiplication is commutative. Thus, the integers are a commutative ring. Also, if we take any two integers, call them $a$ and $b$, and their product $a\cdot b=0$, we know that $a$ or $b$ must be $0$. Any ring that possesses this property is called an integral domain. If there exist two nonzero elements, however, whose product is zero we call such elements zero divisors. This thesis focuses on factorization in commutative rings with zero divisors. In this work we extend the theory of factorization in commutative rings to polynomial rings with zero divisors. For a commutative ring $R$ with identity and its polynomial extension $R[X]$ the following questions are considered: if one of these rings has a certain factorization property, does the other? If not, what conditions must be in place for the answer to be yes? If there are no suitable conditions, are there counterexamples that demonstrate a polynomial ring can possess one factorization property and not another? Examples are given with respect to the properties of atomicity and ACCP. The central result is a comprehensive characterization of when $R[X]$ is a unique factorization ring.

Title: Generalized injectivity and approximations Author: Serap S¸ahinkaya Department:Algebra Faculty of Mathematics and Physics, Charles University Supervisor: Prof. RNDr. Jan Trlifaj, DSc, Faculty of Mathematics and Physics, Charles University Abstract: Injective modules form a basic class studied in contemporary module theory. One of their generalizations, inspired by tilting theory, is the notion of a cotilting module. While tilting modules behave well with respect to localization, we show that colocalization is the correct approach when comparing the struc- ture of cotilting modules over commutative noetherian rings R with the structure of cotilting modules over their localizations Rm where m runs over the maximal spectrum of R. This is done in Chapter 2 of this Dissertation whose main results were published in the paper [33]. In Chapter 3, we investigate approximation properties of other classic generalizations of injective modules, the Ci- and quasi- injective modules, introduced by Harada et al. Suprisingly, we prove that these classes provide for approximations only in exceptional cases (when all Ci mod- ules are injective, or pure-injective). The Dissertation ends with a set of open problems. Keywords: Commutative noetherian ring, (co)tilting module, (generalized) injec- tive module. 1

Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties. Let R be the polynomial ring Ζ[x1,…,xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal I_h generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi's quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Gröbner basis, which is a useful property. Using the Gröbner basis for Jh, we identify a basis for the quotient R/Jh. We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h>h', we have Ih ⊂ Ih' and Jh ⊂ Jh'. We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety. The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter generalization of Springer varieties, parametrized by a nilpotent operator X and a Hessenberg function h. These varieties were introduced in 1992 by De Mari, Procesi and Shayman. We provide evidence that as h varies, the quotient R/Jh may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.

The thesis compiles my contributions to the tilting theory, mainly in the set- ting of a module category over a commutative ring. We give a classification of tilting classes over an arbitrary commutative ring in terms of data of geometrical flavor - certain filtrations of the Zariski spectrum. This extends and connects the results known previously for the noetherian case, and for Prüfer domains. Also, we show how the classes can be expressed using the local and Čech homology the- ory. For 1-tilting classes, we explicitly construct the associated tilting modules, generalizing constructions of Fuchs and Salce. Furthermore, over any commuta- tive ring we classify the silting classes and modules. Amongst other results, we exhibit new examples of cotilting classes, which are not dual to any tilting classes - a phenomenon specific to non-noetherian rings. 1

The first few chapters of the dissertation will catalogue what is known regarding z-ideals in commutative rings with identity. Some special attention will be paid to z-ideals in function rings to show how the presence of the topological description simplifies z-covers of arbitrary ideals. Conditions in an f-ring that ensure that the sum of z-ideals is a z-ideal will be given. In the latter part of the dissertation I will generalise a result in higher order z-ideals and introduce a notion of higher order d-ideals / Mathematical Sciences / M. Sc. (Mathematics)