RESULTS ON N-ABSORBING IDEALS OF COMMUTATIVE RINGS by Alison Elaine Becker The University of Wisconsin-Milwaukee, 2015 Under the Supervision of Dr. Allen Bell Let R be a commutative ring with 1 6= 0. In his paper On 2-absorbing ideals of commutative rings, Ayman Badawi introduces a generalization of prime ideals called 2-absorbing ideals, and this idea is further generalized in a paper by Ander...

We fix once and for all a positive integer n. Unless we say otherwise, Ω will be an open subset of R. We let N; P; Z; Q; R; C be the set of nonnegative integers; the set of positive integers; the ring of integers; the field of rational numbers; the field of real numbers; and the field of complex numbers, respectively. Whenever a ∈ R and 0 < r <∞ we let U(a, r) = {x ∈ R : |x− a| < r} and B(a, r)...

Since rt is a finitely generated group, the space Hom0r, G) is a real analytic variety whenever G is a connected Lie group, and is a real affine algebraic variety whenever G is a linear algebraic group over R [3, 18, 27, 32]. The group G acts on Hom(Tr, G) by conjugation and the orbit space will be denoted by Horn(n, G)/G. Geometrically, the G-orbits on Hom(Tr, G) parametrize equivalence classe...

For relatively prime positive integers u0 and r, we consider the arithmetic progression {uk := u0 + kr} n k=0 . We obtain a new lower bound on Ln := lcm{u0, u1, . . . , un}, the least common multiple of the sequence {uk} n k=0 . In particular, we show that Ln ≥ u0r(r + 1) whenever α ≥ 1 and n ≥ 2αr; this result improves the best previous bound for all but three choices of α, r ≥ 2. We sharpen t...

A double sequence x = {xk,l} of points in R is slowly oscillating if for any given ε > 0, there exist α = α(ε) > 0, δ = δ(ε) > 0, and N = N(ε) such that |xk,l−xs,t| < ε whenever k, l ≥ N(ε) and k ≤ s ≤ (1 + α)k, l ≤ t ≤ (1 + δ)l. We study continuity type properties of factorable double functions defined on a double subset A × A of R into R, and obtain interesting results related to uniform cont...

Let R be a Noetherian standard N d-graded ring and M,N finitely generated, N d-graded R-modules. Let I1, . . . , Is be finitely many homogeneous ideals of R. We show that there exist linear functions f, g : Ns → Nd such that the associated primes over R0 of [Exti(N,M/I1 1 · · · I ns s M)]m and [Tori(N,M/I n1 1 · · · I ns s M)]m are stable whenever m ∈ N d satisfies m ≥ f(n1, . . . , ns) and m ≥...

Over the years, different types of hyperideals have been introduced in order to let us fully realize structures hyperrings general. The aim this research work is define and characterize a new class Krasner $(m,n)$-hyperring that we call n-ary $J$-hyperideals. A proper hyperideal $Q$ with scalar identity $1_R$ said be an $J$-hyperideal if whenever $x_1^n \in R$ such $g(x_1^n) Q$ $x_i \notin J_{(...

Let G be a finite group and let r be the maximum of the p-ranks of G for all primes p dividing the order to G. There exist positive integers m and n such that the exponents of Hn(G,Z) and Hm(G, Z) are greater than Suppose that G is a finite group. It has long been known that the exponent of Hn(G, Z) divides |G|, the order of G, for all n > 0. Beyond this fact very little has been established co...

The codegree density γ(F ) of an r-graph F is the largest number γ such that there are F -free r-graphs G on n vertices such that every set of r−1 vertices is contained in at least (γ−o(1))n edges. When F = PG2(2) is the Fano plane Mubayi showed that γ(F ) = 1/2. This paper studies γ(PGm(q)) for further values of m and q. In particular we have an upper bound γ(PGm(q)) ≤ 1− 1/m for any projectiv...