1
vote
1answer
32 views

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. ...
2
votes
1answer
48 views

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) ...
1
vote
0answers
47 views

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
3
votes
1answer
26 views

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 ...
1
vote
1answer
47 views

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. ...
5
votes
1answer
97 views

Difference between graded ring and graded algebra

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 ...
1
vote
1answer
58 views

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 ...
1
vote
1answer
42 views

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 ...
7
votes
1answer
99 views

Names for specific rings

Is there a name for a ring $A$ (commutative with $1 \neq 0$) all of whose ideals are radical? Is there a name for a ring $A$ (commutative with $1 \neq 0$) such that for each $x \in A$, there is $n ...
2
votes
1answer
176 views

What does “Hauptidealsatz” mean in “Krull's Hauptidealsatz”?

What does "Hauptidealsatz" mean in "Krull's Hauptidealsatz"? Thank you very much.
1
vote
1answer
209 views

dimension of an ideal (definition)

Let $A$ be a commutative ring and $I$ an ideal. When we refer to the "dimension" of $I$, what exactly do we mean? Is it the Krull dimension of $A/I$? In particular, i am trying to understand the ...
5
votes
1answer
70 views

A prime poset of ideals

Let $A$ be a ring (commutative unital), and $\mathcal I$ be a nonempty family of proper ideals of $A$. I will say that $\mathcal I$ has property $\dagger$ if for any $\mathfrak a\in\mathcal I$ and ...
12
votes
3answers
500 views

What are rational integer coefficients?

I have a question about the following excerpt from Atiyah-Macdonald (page 30): “A ring $A$ is said to be finitely generated if it is finitely generated as a $\mathbb Z$-algebra. This means ...
1
vote
2answers
91 views

Quibble with terminology

Proposition 5.15 on page 63 in Atiyah-Macdonald goes as follows: Let $A \subset B$ be integral domains, $A$ integrally closed, and let $x \in B$ be integral over an ideal $ \mathfrak a$ of $A$. Then ...
1
vote
1answer
97 views

Is there a name for this ideal constructed in terms of two submodules?

If $M$ is an $R$-module and $M_1, M_2$ are submodules of $M$, then one can construct the ideal $\{ r \in R \mid rM_2 \subseteq M_1 \}$, which is denoted $(M_1 : M_2)$. Does this construction have a ...
3
votes
0answers
174 views

What's the origin of the terminology “Normalization” in commutative algebra?

Since the terminology "normal", "normalized", etc has different meanings in mathematics (some geometric in flavor, like when referring to perpendicularity) and I just read in Eisenbud's book on ...
11
votes
2answers
293 views

Preimaging units to units

I'm interested in (unity-preserving) homomorphisms $f: S \to T$ between (commutative, with-unity) rings $S$ and $T$ so that if $f(x)$ is a unit, then $x$ was a unit to start with. For example, an ...
3
votes
2answers
474 views

Unital homomorphism

What is a unital homomorphism? Why are they important?