# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### I don't get ring theory. What am I doing wrong? Please help. [closed]

Please allow me to ramble a bit. All my rings are commutative with $1$. I've done two semester's worth of commutative algebra; in particular, a 3rd year undergraduate subject called "Rings, Modules ...
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### Every irreducible submodule is primary

Sea $R$ un anillo conmutativo con identidad noetheriano y $M$ un $R$-módulo finitamente generado. $N$ es un submódulo propio de $M$. Entonces si $N$ es irreducible implica que $N$ es primario. ...
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### When is $X_1^{a_1} \cdots X_n^{a_n}-1$ irreducible?

Let $F$ be a field, and $a_1, ... , a_n \geq 1$ integers. When is the polynomial $$f = X_1^{a_1} \cdots X_n^{a_n}-1$$ irreducible in $F[X_1, ... ,X_n]$? I believe this should be the case if and ...
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### $(x_1, …, x_k)$ is prime in $R[x_1, …, x_n]$ if $R$ is an integral domain

Let $R$ be an integral domain. I need to prove that $\forall k = 1, ..., n \ \ \ (x_1, ..., x_k)$ is prime in $R[x_1, ..., x_n]$. I managed to do it for $k = 1$. Let $f, g \in R[x_1, ..., x_n]$. Then ...
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### Does localization commute with taking radicals?

Let $A$ be a ring, $S\subset A$ a multiplicative set, and $I\subset A$ an ideal not intersecting $S$. For any ideal $J$, let $r(J)$ denote the radical of $J$. Is $S^{-1}r(I) = r(S^{-1}I)$? ...
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### Proof of Krull's intersection theorem with Taylor expansion

I took a commutative algebra course last semester (with Kaplansky's book), and I've learned about Krull's intersection theorem. In the course, we proved it without using Artin-Rees Lemma. I heard that ...
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### Basic fibres for monomials

The following definitions are from Irena Peeva's book Graded Syzygies. Let S = k[x$_1$,...,x$_n$], the set of all monomials in S of multidegree $\alpha$ is called the fibre of $\alpha$. We denote gcd(...
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### If $p$ is a prime ideal then $p[X]$ is a prime ideal

If $Z$ is a ring and $p$ is a prime ideal of $Z$ then $p[X]$ is a prime ideal of $Z[X]$. Is it true or false? I believe that it is true and I try to prove it like that: Take $f(x)\in p[X]$ and ...
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### Group action on algebra over a field defined on generators

Suppose $G$ is a group and $A$ is a finitely generated algebra over a field $\mathbb{k}$. Let $X=\{x_1,...,x_n\}$ be a set of generators for $A$, and suppose $G$ acts on $X$. Is this enough to define ...
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### Shouldn't $t^n : \mathbb{A}^1 \rightarrow \mathbb{A}^1$ ramifies at $0$?

Yo, this is probably the stupidest question ever that I've asked here. Let $$\varphi: \mathbb{A}^1 \rightarrow \mathbb{A}^1$$ be the map of schemes (over a field $k$) such that $\varphi (x) = x^n$. ...
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### Maximal ideal in a local artinian ring.

I know that an artinian ring $A$ is the union of its units and its zero-divisors. So every non-zero-divisor is an unit. I also know that in a local ring every element which is out from the maximal ...
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### $R$ integral domain, $P$ projective and injective module $\implies P=0$ or $R$ is field of fractions [duplicate]

I'm having difficulties proving the following: Let $R$ be an integral domain and $P$ a projective and injective $R$-module. Show that $P=0$ or $R=Q(R)$, where $Q(R)$ denotes the field of fractions of ...
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### Irreducible elements for a commutative ring that is not an integral domain

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral ...
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### Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
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### How do I find the ideal $I+J$ and quotient $R/(I+J)$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
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### A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. [closed]

I expect that the following result is true, but i can't prove it. A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. I need some help to prove this....
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### Zariski tangent vectors, dual numbers

Let $k$ be a field, $A$ be a Noetherian local $k$-algebra, $m$ its maximal ideal, and an isomorphism $i:A/m \to k$ . Let $v:m/m^2 \to k$ be a $k$-linear map (i.e. a Zariski tangent vector). I believe ...
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### Example of an monomial ideal that is weakly reverse lexicographic but not reverse lexicographic

We are looking at a paper titled "Generic Ideals and Moreno-Socias Conjecture" by Edith Aguirre, et al. In the paper they state that an ideal which is reverse lexicographic is also weakly ...
Let $A \subseteq B$ be rings (comm. with unity). I am struggling to see why the following equivalence holds for $B$ interpreted as a $A$-Algebra: $A \rightarrow B$ is of finite type and $A\... 1answer 50 views ### How do you find the free resolution of the module$M$and of$F/M$where$F=(K[x,y])^3$?$M$is a module generated by $$f_1=(xy,y,x), f_2=(x^2+x,y+x^2,y), f_3=(-y,x,y),f_4=(x^2,x,y).$$ We're to use the lex ordering with$x<y$and$e_1>e_2>e_3$, where terms are given preference ... 1answer 65 views ### Help finding an article [closed] Hello Recently I have been studying algebra and am in search of the following paper : Kac, V. G. Classification of simple$Z\$-graded Lie superalgebras and simple Jordan superalgebras. Comm. Algebra 5 ...
We have $$I=\left(x^2+2y^2-3,y(x-y),y(y+1)(y-1)\right)\subset\mathbb{C}[x,y]$$ and I would like to decompose it as intersection of simpler ideals. How could I proceed? For example, in this ...