# Tagged Questions

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### Is the colimit of finite tensor products a tensor product?

Let $(R_\lambda)_{\lambda\in\Lambda}$ be a family of $A$-algebras. Atiyah & MacDonald defines the "tensor product" of the family as the direct limit of the tensor product of finite subfamilies. ...
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### In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
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### Name of a certain type of rings

What is the name given to (if there exists any) commutative rings $R$ with identity such that $R/(a)$ is finite for every non-zero $a\in R$ Thanks a lot
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### Terminology regarding property of ideals

Is there a name for a property that only needs to be checked for either prime or maximal ideals in order to show that it holds for all ideals? An example would be being a principal ideal for which ...
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### Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
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Wikipedia says that a graded $A$-algebra is just a graded $A$-module that is also a graded ring. Question: when one says then "finitely generated graded $A$-algebra", does one mean that every element ...
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### why are these rings called fibres?

This question is self-contained. In the book "Monomial Ideals", by Herzog and Hibi, p. 45, we have the following definition: Definition: Let $K$ be a field. A one-parameter flat family of ...
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### What is the name of this factor-algebra?

In the polynomial algebra $k[x_1,x_2,\ldots, x_n]$ consider an ideal $I$ generated by the polynomials of the form $x_i^k-x_i$, $i=1 \ldots n$ and $k=2,3,\ldots.$ Consider the quotient algebra ...