# Tagged Questions

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### In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
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### Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
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### Why are groups “abelian” but rings “commutative”?

I have never seen, in any text, a ring whose multiplication is commutative being called an "abelian ring", even though this would make perfect sense, because this term would necessarily refer to ...
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### History of the terms “prime” and “irreducible” in Ring Theory.

In ring theory, a nonzero, nonunit element $p$ of a integral domain is called irreducible if $p=ab$ implies that exactly one of $a$ and $b$ is a unit, and it's called prime if $p\mid ab$ implies that ...
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### Ring $R$ as $R[x]$-module

My professor mentioned some interesting examples of modules, giving as an example the following two: $R$ as an $R[x]$-module, in which multiplication by $x$ was taken to be evaluation under a fixed ...
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### Terminology regarding property of ideals

Is there a name for a property that only needs to be checked for either prime or maximal ideals in order to show that it holds for all ideals? An example would be being a principal ideal for which ...
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### What is the name for a ring without nilpotent elements?

Let $n,m$ be positive integers. What is the name for a ring $A$ that satisfies the two conditions : $1)$ The ring $A$ is isomorphic to ${\Bbb R}^{n}\times{\Bbb C}^{m}$. $2)$ For every nonzero ...
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### Is there a special name for ring homomorphisms $f : R \rightarrow S$ with $f^*(C(R)) \subseteq C(S)$?

Edit. For some reason, I called the functor $F$ described below a full functor as opposed to a faithful functor. The problem has now been corrected. For any ring $R$, let $C(R)$ denote the center of ...
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Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, ... 3answers 267 views ### Is there a name for this ring-like object? Let$S$be an abelian group under an operation denoted by$+$. Suppose further that$S$is closed under a commutative, associative law of multiplication denoted by$\cdot$. Say that$\cdot$... 1answer 270 views ### What does “quotient-ring” mean? I am reading a paper about rings (http://malaschonok.narod.ru/publ/ma01.ps, page 3). In this paper the term "quotient-ring" appeared. What is a quotient-ring? (Note: The text in the original ... 0answers 32 views ### Terminology: Subrings with the property that an element is invertible iff it is invertible in the larger ring. [duplicate] Possible Duplicate: Is there a term for an “inverse-closed” subring of a ring? This is a question about terminology. Is there a standard name for a subring$A \subset B$that ... 0answers 51 views ### What are 'symmetric (or 'new age') ring structures'? The title says it: what are 'symmetric (or 'new age') ring structures'? The phrase was found in: Frank Quinn, Contributions to a Science of Contemporary Mathematics (Draft October 4, 2011), p 74 ... 1answer 309 views ### Ideals in non-associative rings and the identity$(xy)z=y(zx)$. I have come across this paper. The authors prove that magmas satisfying the identity $$(xy)z=y(zx)\tag1$$ are nearly both associative and commutative. To be precise, they show that in such magmas, ... 1answer 270 views ### What are the rings in which left and right zero divisors coincide called? A unital ring$R$is reversible iff$ab=0\implies ba=0.$This condition implies the following one. If$a\in R$is a left-zero divisor, then$a$is also a right-zero divisor. And the other way ... 2answers 384 views ### What is an element of a rng called which is not the product of any elements? Let$R$be a non-unital ring. Let$F:R\times R\longrightarrow R$be a function given by the formula$F(x,y)=xy.$Let$r\not\in\operatorname{im}(F).$Such elements can exists, for example$2\in ...
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...