WebFeb 17, 2024 · commutative algebra. Even though the definitions of the Noetherian and Artinian properties are dual to each other, it turns out that the Noetherian condition is more important. For instance, we have already seen that every Artinian ring is Noetherian. In this post, we will prove more properties of Noetherian modules and rings. WebAn algebra is strongly separable if and only if its trace form is nondegenerate, thus making the algebra into a particular kind of Frobenius algebra called a symmetric algebra (not to be confused with the symmetric algebra arising as the quotient of the tensor algebra). If K is commutative, A is a finitely generated projective separable K ...
Non finitely-generated subalgebra of a finitely-generated algebra
WebLet $k$ be a field and $A$ a finitely generated algebra over $k$ that doesn't have zero divisors. Why is the integral closure of $A$ a finitely generated module over ... The polynomial algebra K[x1,...,xn ] is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.The field E = K(t) of rational functions in one variable over an infinite field K is not a finitely generated algebra over K. On the other hand, E is generated over K by a … See more In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A such that every element of A … See more • Finitely generated module • Finitely generated field extension • Artin–Tate lemma See more • A homomorphic image of a finitely generated algebra is itself finitely generated. However, a similar property for subalgebras does not hold in general. • Hilbert's basis theorem: if A is a finitely generated commutative algebra over a Noetherian ring then … See more the old bell
Integral element - Wikipedia
Web10.35. Jacobson rings. Let be a ring. The closed points of are the maximal ideals of . Often rings which occur naturally in algebraic geometry have lots of maximal ideals. For example finite type algebras over a field or over . We will show that these are examples of Jacobson rings. Definition 10.35.1. Web31. No, being finitely generated as an algebra is generally not as strong as being finitely generated as a module. Being finitely generated as an algebra means that there is … WebAug 31, 2024 · In other words, if k k is a perfect field, there is no difference between a separable algebra over k k and a finite-dimensional semisimple algebra over k k. ... If a separable algebra A A is also projective as a module over k k, it must be finitely generated as a k k-module. For more details see DeMeyer-Ingraham. mickey loungefly mini backpack