UC Riverside

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}).