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For a domain A containing a field k with tr.degkA < ∞, we define a new transcendence degree of A with respect to k, which is denoted by tdkA. By using this, we generalize the theorem that for every affine domain A over a field k it holds that dim A = tr.degkA. For example, we show that if A is a quasi-local domain containing a field k with dim A = tdkA < ∞, then for every Noetherian local k-subalgebra R of A it holds that dim R = tdkR. Moreover we also generalize the theorem due to Gilmer, Nashier and Nichols.

In this paper, we determine some sufficient conditions for an $A\,+\,XB\left[ X \right]$ domain to be an $\text{HFD}$. As a consequencewe give new examples of $\text{HFDs}$ of the type $A\,+\,XB\left[ X \right]$.

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The transcendence degree of an integral domain over a subfield and the dimension of the domain

Examples of Half-Factorial Domains

On Complete Intersections and Their Hilbert Functions

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3 results

The transcendence degree of an integral domain over a subfield and the dimension of the domain

Examples of Half-Factorial Domains

On Complete Intersections and Their Hilbert Functions