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Let $R$ be a commutative rng, i.e. a commutative ring without an identity element. Does $R$ still have the Invariant Basis Number (IBN) property? Recall that a ring is said to have the IBN prope …

First of all, I apologize if this is too elementary for MO. I am reposting this from MSE where I asked this question two days ago (link to MSE post). I've been wondering about the following thing. L …

Browsing through my notes on Artin rings, I have realized that I don't know an example for this and I wasn't able to google anything relevant. What is an example for a commutative non-Noetherian loca …

Going through my Commutative Algebra notes I found out that I don't know the answer to this question. Let $A$ be a commutative ring with unit and let $f(T):=\sum_{k=0}^\infty a_k T^k \in A[[T]]$ be a …

Let $A$ be a commutative ring with unity. Denote $X:=Spec(A)$. $X$ is connected if and only if $A$ cannot be written as a Cartesian product $A_1\times A_2$ for nontrivial $A_1$ and $A_2$. So, that g …

I am reading Shurygin's survey "Smooth Manifolds over Local Algebras and Weil Bundles" (Journal of Math. Sciences, Vol. 108, No. 2, 2002) and it mentions the following basic fact which I don't quite u …

A local ring $(A,\mathfrak{m})$ is regular iff the minimal number of generators of $\mathfrak{m}$ equals the Krull dimension of $A$. But Artinian rings have Krull dimension $0$, i.e. $\mathfrak{m}=0$, …

All rings are assumed commutative and unital. Some context (feel free to skip right to the questions below). I am trying to understand to what extent the property "being prime" for an ideal $I$ is int …

No, for rather trivial reasons. Consider the polynomial $f(X) = \varepsilon X - 1$ with $\varepsilon^2 = 0$, $\varepsilon \neq 0$, in $R = M_2(\mathbb{C})$. Then a root of $f(X)$ would mean that $\var …

I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the st …

In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$). Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its g …

Consider the ring $\mathbb{C}[[X,Y]]$ and its subring $\mathbb{C}[[X+iY]]$, where $i=\sqrt{-1}$. One can show that $f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$ lies in $\mathbb{C}[[X+iY]]$ iff $u$ an …

I would suggest Altman and Kleiman's "A Term of Commutative Algebra". As far as I know, the most recent version can be found at https://dspace.mit.edu/handle/1721.1/116075.2. Quoting from the preface: …

I will address the question in the case $f(z) = (-1)^z = \exp(z \log(-1))$ and $A$ is a finite-dimensional commutative (associative unital) $\mathbb{R}$-algebra, where $\log(-1)$ is a suitable choice …

Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group …