The goal of this paper is to continue building lattice theory by generalizing known results from commutative rings. Our focus will be results concerning strong Mori lattices as Mori lattices have been rather developed already in [7], [22], and others sources.

Our first main objective will be to obtain more results concerning quotient field lattices, which have been significantly developed in [6], [7], and [22]. Most notably, the quotient eld lattice version of the Nagata Theorem will be proven, following the approach of Chang Hwan Park and Mi Hee Park in [23]. This result had not yet been generalized to quotient field lattices. Also, some results concerning composition series will receive lattice versions, generalizing results from [23] as well as [15].

Some concepts from commutative ring theory that have not yet been applied to lattices will be introduced, such as certain star operations, particularly the w-operation. This will allow us to develop strong Mori lattices and begin to characterize them, working toward achieving lattice

versions of results from [27] and [28]. We will also introduce a lattice version of w-invertibility and obtain lattice versions of basic results. This will give a brief characterization of strong Mori lattices and lay the groundwork for further, more detailed characterizations as discussed in the final section.

Let R be a locally quasi-unmixed domain, a, b₁,&hellip, b_n an

asymptotic sequence in R, I=(a, b₁,&hellip, b_n)R, and

S=R[b₁/a,&hellip, b_n/a]=R[I/a]. Then S is a locally quasi-unmixed

domain, a, b₁/a,&hellip, b_n/a is an asymptotic sequence in S, and

there is a one-to-one correspondence between the asymptotic primes

Ass_R(R/(Iⁿ)_a) of I and the asymptotic primes Ass_S(S/(aSⁿ)_a)

of aS=IS. Moreover, if a, b₁,&hellip, b_n is an R-sequence,

then that one-to-one correspondence extends between Ass_R(R/I) and

Ass_S(S/aS).

We give a sufficient condition for the monoidal transform

S to be a unique factorization domain, or a Krull domain whose class

group is torsion, finite, or finite cyclic. As a corollary, we give a

necessary and sufficient condition for R and its monoidal transform

to have the same class group.

In the case that R is a unique factorization domain, we examine the

height-one prime ideals of S to determine how far S is from unique

factorization. In Section 3.2, a complete description is given of which

height-one prime ideals P of S are principal or have a prinicpal primary

ideal in the case that the contraction to R of P, which we will call p, has height one. In Section 3.3, we show that if the

prime factors of a satisfy a mild condition, we may give a similar

description in the case that p has height greater than one. We give a necessary and

sufficient condition for S to be a Krull domain with finite cyclic

class group in the case that a is a power of a prime element, and we show

that this holds for the Rees ring R[1/t,It] as a monoidal transform

over R[1/t] as well. Furthermore, if a is a power of a prime element,

we show that if Rad(I) is not prime and p is a height-one prime ideal

of R contained in at least one but not all asymptotic prime divisors of I, then

the height-one prime ideal P of S, which can be represented as pR[1/a] intersected with S, has no principal primary

ideal.

In this paper, we study the Cohen-Macaulay property of a general commutative ring with unity defined by Hamilton and Marley. We give sufficient conditions on pullback constructions, fixed rings, and normal monoid rings to all be Cohen-Macaulay in this sense. We also exhibit a class of quasi-local rings where the top Čech cohomology module with respect to a sequence generating the maximal ideal up to radical is non-vanishing.

In this dissertation, we study three recent generalizations of unique factorization; the almost Schreier property, the inside factorial property, and the IDPF property. Let R be an integral domain and let p be a nonzero element of R. Then, p is said to be almost primal if whenever p divides xy, there exists a positive integer k and a, b in R such that p^k=ab with a | x^k and b | y^k. R is said to be almost Schreier if every nonzero element of R is almost primal. Given an M-graded domain R=(bigoplus_{m in M} R_m), where M is a torsion-free, commutative, cancellative monoid, we classify when R is almost Schreier under the assumption that the extension from R to its integral closure is root. We then specialize to the case of commutative semigroup rings and show that if R[M] to its integral closure is a root extension, then R[M] is almost Schreier if and only if R is an almost Schreier domain and M is an almost Schreier monoid.

Let D_n(a) denote the set of non-associate irreducible divisors of a^n. R is said to be IDPF, if for every nonzero, nonunit element a of R, the ascending chain D_1(a) subset D_2(a) subset ... stabilizes on a finite set. Also, a monoid H is inside factorial if there exists a divisor homomorphism phi : D -> H from a factorial monoid D such that for any x in H there is a positive integer n with x^n in the image of D under phi. R is inside factorial if its multiplicative monoid R-{0} is inside factorial. Continuing our investigation of semigroup rings, we prove that no proper numerical semigroup ring R[S] of characteristic zero is IDPF. Let R be an order in any quadratic integer ring and let n be the least positive integer in the conductor ideal. We tie the IDPF, inside factorial, and the almost Schreier properties together by proving that R[X] is IDPF if and only if R[X] is almost Schreier if and only if R[X] is inside factorial if and only if every prime divisor of n also divides the discriminant of Q(sqrt{d}).

The goal of this dissertation is to investigate the properties of semistar operations and integral domains in terms of their natural partial ordering. We mainly focus on the relationships between different classes of integral domains in terms of semistar operations.