# Questions tagged [commutative-algebra]

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

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### Observations on the associativity of the tensor product between two modules.

I have a little question about of the associativity of the tensor products of two modules. If $M, N, P$ are $A$-modules, we know that $(M \otimes N) \ P \cong M \otimes N \otimes P$ If i want to prove ...
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### Is this quotient of a polynomial ring local?

Let $k$ be a field, $R = k[X_1, \ldots, X_n]$ and $I = \langle X_1, \ldots, X_n\rangle$. I am trying to prove that the quotient ring $R/I^r$, $r\in\mathbb{N}$ is local, i.e., has only one maximal ...
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### Name for algebra which is commutative up a group action

I am wondering if there is a name for an algebra which is commutative up to some group action. To be more concrete, assume $A= \bigoplus A_n$ is a graded algebra, so $A_n \cdot A_m \subset A_{n+m}$, ...
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### In the proof of extension of a homomorphism, proposition 5.23 of Atiyah-Macdonald, Introduction to commutative algebra.

I don’t understand the part of the proof of prop5.23 of Atiyah-Macdonald, Introduction to commutative algebra. The author separates the proof into two cases, the case that x is transcendental or the ...
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### Minimal elements under division in a ring

Let $p$ be an element of a commutative ring with unity. The following definition is natural: $p$ is minimal under division if its only divisors (up to equivalence) are $1$ and itself. That is, for ...
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### A theorem of Bertini and regular sequence.

In Exercise 10.4 (p.243) in Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud, it writes Let $a,b$ be a regular sequence in a domain $R$, and let $S=R[x]$ be the polynomial ...
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### Integral domain containing non-prime irreducibles (hence not a UFD) where all factorizations into irreducibles are unique

An integral domain is called a UFD if (1) every non-zero non-unit element factors into irreducibles, and (2) every element that factors into irreducibles does so uniquely (up to units and order). It ...
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### Why are the underlying polynomials of a rational matrix function commutative?

I am currently reading the book "The Theory of Matrices" where one chapter deals with the extension of scalar-valued functions to quadratic matrices. In the book, a matrix function is ...
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