# Tagged Questions

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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### If $B$ is an ideal of $A$ then $B[x]$ is an ideal of $A[x]$ - what's wrong with my proof?

This is exercise E.2 from chapter 24 of Pinter's A Book of Abstract Algebra: If $B$ is an ideal of $A$, $B[x]$ is not necessarily an ideal of $A[x]$. Give an example to prove this contention. It ...
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### Mal'cev condition for variety of rings generated by finite fields to be arithmetical

This is an exercise of Burris & Sankappanavar (Universal Algebra), Chapter II, section 12. It asks to prove that, if $V$ is a variety of rings generated by finitely many finite fields, then $V$ is ...
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### Minimal prime ideals of the ring of continuous functions

Let $X$ be a topological space. Are there any conditions on $X$ which guarantee that that the minimal prime ideals of $C(X)$, the ring of real-valued continuous functions on $X$, have a nice ...
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### Ring->module->$R$-algebra, Field->Vectorspace->algebra

I haven't done any mathematics for a long time, and I have forgotten some things. I want to try to remember some of the words and how they interact. A module is a 'vectorspace over a ring' rather ...
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### Finitely generated module: terminology.

What's the meaning of the expression: $S$ is a subring of $\mathbb{C}$ finitely generated as $\mathbb{Z}$-module? Maybe that the additive group of the ring $S$ is a finitely generated abelian group? ...
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### assuring factorization for R[x] when R is a UFD

I wanted to ask, suppose the ring $R$ is a UFD (Unique factorization domain) and I look at $R[x]$, the ring of polynomials over $R$. I wanted to know, how can I assure that when I have some polynomial ...
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### If the ring $M_n(D)$ is a $k$-algebra, is $D$ a $k$-algebra?

Let $k$ be a field, let $D$ be a division ring. Assume the matrix ring $A = M_n(D)$ is endowed with some $k$-algebra structure compatible with its ring structure (namely, for all $\lambda \in k$ and ...
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### $F(X)$ as a subfield of $F((X))$ of formal Laurent series

$F(X)$ is a subfield of $F((X))$ by considering the Laurent expansion of rational functions at the origin. So what is indeed the degree of this field extension $F((X))/F(X)$? Or this is an infinite ...
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### How do I see that $a = bu$ for some unit $u$? [duplicate]

Suppose elements $a$ and $b$ in a domain satisfy $a \mid b$ and $b \mid a$. How do I see that $a = bu$ for some unit $u$?
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### Examples of $R$-modules $X$ such that $(X \setminus TX) \cup \{0\}$ isn't a submodule.

Work over an ambient commutative ring with unity. Given a module $X$, write $TX$ for its submodule of torsion elements. Suppose we want to find the "submodule" of torsion-free elements of $X$. So ...
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### $a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$? [closed]

As the question title suggests, how do I see that $a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$?
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### Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$? [closed]

Suppose that $a$, $b \in \mathbb{Z}[i]$ satisfy $a \mid b$ and $b \mid a$. Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$?
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### A doubt about the correspondence theorem.

Let $f$ be a ring homomorphism from $R$ onto $R_1$. Then there is a one one correspondence between the set of all ideals of $R_1$ and the set of all ideals of $R$ that contain the kernel. Now what ...
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### Idempotents central or not?

Let $R$ be a nil-clean ring with unity such that $R/J(R)$ is reduced, where $J(R)$ is the Jacobson radical of $R$. Is it true that $R$ is abelian, i.e. the idempotents are central? (By nil-clean I ...
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### How does restriction of scalars interact with tensor products?

Say that we have a morphism of commutative rings $f: R \to S$. Does the restriction of scalars functor $f^*: S \text{Mod} \to R \text{Mod}$ commute with tensor products? In other words, I would like ...
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### Intuitive reasons of ring modulo maximal ideal or prime ideal

Are there any intuitive reasons that can help us remember that $R/I$ is a field iff $I$ is a maximal ideal; $R/I$ is an integral domain iff $I$ is a prime ideal? (I can understand the proof, but have ...
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### What do we mean by a ring and what is ringlike about it? [duplicate]

I see that a ring is a triple $(R,\cdot,+)$. I am confused by the terms abelian group and semigroup. Does this mean for $x \in R$ and $y \in R$, $x \cdot y$ and $x+y$ are defined? If so, how is this ...
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### Notation for the set of zero divisors in a ring

If $R$ is a nonzero ring with identity then I have seen the group of units denoted by $R^{\times}$ or possibly $R^*$ in some texts. In a classical ring there is a trichotomy which declares each ...
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### Dimension under integral local homomorphism

Let $f:(R,m) \to (S,n)$ be an integral local homomorphism. Let $p$ be a prime ideal of $R$ not equal to $m$. I want to know if one can claim $\dim S/f(p)S\neq0$. This is true when $(f(p)S)^c=p$, ...
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### A monomorphism from an $R$-module to $R$

Let $R$ be a commutative ring with unity possessing an element $r$ in the singular ideal $Z(R)=$ the set of elements whose annihilators are essential in the module $R_R$, and let $M$ be a faithful $R$...
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### Meaning of “scaled topology” on a ring

Suppose that $(A,\tau)$ is a DVR with a topology (which is not the metric topology) with field of fractions $K$. In a conference the speaker said that for any $b\in K^\times$ one can define the "...
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### Can we continually factorize an expression like $x+y$?

I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As ...
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### $R/(IJ)$ is reduced $\Rightarrow IJ = I \cap J$ for ideals $I,J$ of a commutative ring $R$

This is exercise $4.6$ on page $154$ of the textbook Algebra: Chapter $0$ (authored by P. Aluffi): Let $I,J$ be ideals of a commutative ring $R$. Assume that $R/(IJ)$ is reduced (that is, it has ...
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### Generalizing concept of content of a polynomial to commutative rings [duplicate]

Let $A$ be a commutative ring with identity. Let $f,g\in A [x]$. Let $I_1,I_2, J$ be the ideals generated by the coefficients of $f,g,fg$ respectively. Must $J$ be equal to $I_1 I_2$ ? It is an ...
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### In abstract algebra, what is an intuitive explanation for a field?

Wikipedia has the following to say about fields. In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or ...
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### Cardinality of the base of a ring of sets

Concretely, my question is: What is the size of the minimal base of a ring of sets? In my understanding the base is the set of elements that can be used to "deduce the rest". E.g., if I have a ring ...
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