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Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by so …

Mnemonic: $\quad M=IM \Rightarrow m=im$ The version of Nakayama described: If $I$ is an arbitrary ideal of an arbitrary commutative ring $A$ and if a finitely generated module $M$ satisfies $M=IM$, …

Dear Tim, on page 31 they consider a ring $A$ and two $A$- algebras defined by their structural ring morphisms $f:A\to B$ and $g:A\to C$. They then define the tensor product as a ring $D=B\otimes _A C …

Since your profile says you are interested in Algebraic Geometry, here are geometric considerations that might appeal to you. Consider a projective module $P$ of finite type over a commutative ring $ …

Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$ My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known? …

Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following. Let's st …

Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory …

No, here is an example of a morphism $f:X\to Y$ which is not affine although $X$ is affine. Take $X=\mathbb A^2_k$, the affine plane over the field $k$ and for $Y$ the notorious plane with origin do …

Dear CJD, if you are still interested in your problem, already solved three weeks ago by Anton, here is another point of view. Let $M:A^m \to A^n$ be injective. Let $B=\mathbb Z [\ldots,m_{ij},\ldots …

A totally ordered finite set $\quad \mathcal P_0 \varsubsetneq \mathcal P_1\varsubsetneq \dots \mathcal \varsubsetneq \mathcal P_n \quad$ of prime ideals of a ring $A$ is said to be a chain of lengt …

Dear roger123, let $R$ be a commutative ring and $M$ an $R$-module ( which I do not suppose finitely generated). In order to minimize the risk of misunderstandings, allow me to introduce the followin …

What I find intriguing is that the Nullstellensatz is underappreciated in the sense that many people appeal to a variation of it without saying (or realizing) they do. For example, Hadamard's lemma …

This is a very interesting question related to the Hilbert scheme $Hilb^n(\mathbb A^d)$ classifying $n$ points in affine space $\mathbb A^d$. I don't think there is a classification but there is an es …

What is the fraction field $K$ of the domain $\mathbb Z[[X]]$? It is strictly smaller than the field of Laurent series $L=\operatorname {Frac}\mathbb Q[[X]]$, since $\sum_{i\geq 0}\frac {X^i}{i!}\in …

Dear liu, 1) If $A$ is a domain in which every finitely generated ideal is principal, then a module over $A$ is flat iff it is torsion free (Bourbaki, Comm.Alg.,I,§2, 4, Prop.3). Of course a PID has …