# t-Reductions of Ideals in Integral Domains

t-Reductions of Ideals in Integral Domains. PhD thesis, King Fahd University of Petroleum and Minerals.

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This Ph.D. thesis traverses two chapters which contribute to the study of multiplicative ideal theoretic properties of integral domains. Let $R$ be an integral domain and $I$ a nonzero ideal of $R$. An ideal $J\subseteq I$ is a $t$-reduction of $I$ if $(JI^{n})_{t}=(I^{n+1})_{t}$ for some integer $n\geq0$. An element $x\in R$ is $t$-integral over $I$ if there is an equation $x^{n}+a_{1}x^{n-1}+...+a_{n-1}x+a_{n}=0$ with $a_{i}\in (I^{i})_{t}$ for $i=1,...,n$. The set of all elements that are $t$-integral over $I$ is called the $t$-integral closure of $I$. The first chapter investigates the $t$-reductions and $t$-integral closure of ideals. Our objective is to establish satisfactory $t$-analogues of well-known results, in the literature, on the integral closure of ideals and its correlation with reductions. Namely, Section 1.2 identifies basic properties of $t$-reductions of ideals and features explicit examples discriminating between the notions of reduction and $t$-reduction. Section 1.3 investigates the concept of $t$-integral closure of ideals, including its correlation with $t$-reductions. Section 1.4 studies the persistence and contraction of $t$-integral closure of ideals under ring homomorphisms. All along the chapter, the main results are illustrated with original examples. An ideal $I$ is $t$-basic if it has no $t$-reduction other than the trivial ones. The second chapter investigates $t$-reductions of ideals in pullback constructions. Section 2.2 examines the correlation between the notions of reduction and $t$-reduction in pseudo-valuation domains. Section 2.3 solves an open problem on whether the finite $t$-basic and $v$-basic ideal properties are distinct. We prove that these two notions coincide in any arbitrary domain. Section 2.4 features the main result, which establishes the transfer of the finite $t$-basic ideal property to pullbacks in line with Fontana-Gabelli's result on Pr\"ufer $v$-Multiplication Domains (P$v$MDs) and Gabelli-Houston's result on $v$-domains. This allows us to enrich the literature with new families of examples, which put the class of domains subject to the finite $t$-basic ideal property strictly between the two classes of $v$-domains and integrally closed domains.