0
votes
1answer
43 views

Explanation about notation

Can anyone explain to me what $C[x^2,x^3]_{(x^2,x^3)}$ means? It is connected with localizations but it is unclear to me what it means exactly.
0
votes
2answers
117 views

Atiyah-MacDonald 5.10

The problem says: Let $f:A\rightarrow B$ be a ring homomorphism and let $f^{*}:\operatorname{Spec}(B)\rightarrow \operatorname{Spec}(A)$ be the mapping associated with $f$. And then comes the ...
3
votes
2answers
112 views

Show that a local ring is equicharacteristic iff it contains a subfield

A local ring $(A,\mathfrak m)$ is equicharacteristic if $\operatorname{char} A=\operatorname{char} \kappa (m)$. Need hints to solve the following question: A local ring is equicharacteristic ...
3
votes
1answer
181 views

What does $R_P$ mean, for a ring $R$ and an ideal $P$?

What does $R_P$ mean, for a ring $R$ and an ideal $P$? This appeared in some notes by a teacher of mine, but he didn't define this notation. He used it as follows: suppose $R$ is a commutative ring, ...
3
votes
2answers
115 views

What letter should I use to denote an ideal?

In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$. By itself, it is hardly surprising - after ...
4
votes
4answers
161 views

Why Does Finitely Generated Mean A Different Thing For Algebras?

I've always wondered why finitely generated modules are of form $$M=Ra_1+\dots+Ra_n$$ while finitely generated algebras have form $$R=k[a_1,\dots, a_n]$$ and finite algebras have form ...
1
vote
1answer
94 views

Notation question for denoting an ideal of a polynomial ring

Let $k$ be a field. Let $q=(x,y^2)$ be an ideal of $k[x,y]$. What exactly does the notation $q=(x,y^2)$ mean, i.e. what kind of elements does $q$ contain? Is it the set of all elements $\alpha x + ...
5
votes
1answer
194 views

Notation in Atiyah - Macdonald

I am now going through some problems in Atiyah - Macdonald Chapter 3. In problems 21 and 23 of chapter 3, they use the notation $A_f$ to mean something I don't know. I have not seen this before. ...
4
votes
1answer
101 views

For a ring of char $p$ where $p>0$ is a prime, what does $R^{1/p}$ mean?

If $R$ is a ring of characteristic $p\gt 0$, what does $R^{1/p}$ mean? I am not sure how to search for it, since I don't know a name for it. From the notation, it seems to be a ring consisting of the ...
2
votes
1answer
104 views

Trouble with notation $I:J^{\infty}$

I am not sure I understand this notation correctly. The definition says, for a ring $R$ with $I,J$ ideals of $R$, we define $I:J^{\infty}=\cup_{i=1}^{\infty} I:J^i$. Now, $I:J$ is the set of elements ...
3
votes
2answers
981 views

Notation for a polynomial ring and formal polynomials

Given that we shouldn't say that "$f(z)$ is a function", shouldn't we also not write "$p \in k[X_1, \ldots, X_n]$ is a polynomial"? Along those lines, I usually write $p(X_1, \ldots, X_n) \in k[X_1, ...