Tagged Questions

1
vote
2answers
28 views

Help to understand the ring of polynomials terminology in $n$ indeterminates

In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner: After the author defines the operations in this ring with a theorem: ...
2
votes
2answers
64 views

Why the terms “unit” and “irreducible”?

I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition Maybe historical reasons? For example, I suppose the second ...
3
votes
1answer
67 views

What do we call this property: there exists $n \geq 2$ such that $x^n=x$

Let $R$ denote a ring and suppose $x \in R$. If $x^2=x$, we call $x$ idempotent. If there exists $n \geq 2$ such that $x^n=0$, we call $x$ nilpotent. Suppose there exists $n \geq 2$ such that $x^n = ...
12
votes
2answers
2k views

What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?

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, ...
10
votes
3answers
206 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 $\cdot$ ...
2
votes
1answer
235 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 ...
2
votes
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 ...
1
vote
0answers
44 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 ...
4
votes
1answer
212 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, ...
10
votes
1answer
204 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 ...
0
votes
0answers
85 views

Banach modules; ambiguous terminology

In the following article, S. Grabiner very often writes "Banach module $A$ finitely dimensional over its radical". What does it mean? Does it mean that we think about the module as a module over its ...
8
votes
2answers
361 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 ...
40
votes
4answers
2k views

Why “characteristic zero” and not “infinite characteristic”?

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 ...
6
votes
1answer
427 views

Why is it called a 'ring', why is it called a 'field'?

The definitions of 'ring' and 'field' are pretty straightforward. For a ring (e.g. integers): addition is commutative $( 1 + 2 = 2 + 1 )$ addition and multiplication are associative $(2 +(2+2)) = ...
9
votes
1answer
204 views

Is there a term for an “inverse-closed” subring of a ring?

I would like to know whether there are established terms for A subring $S$ of a ring $R$ such that $S \cap U(R) = U(S)$; in other words, every element of $S$ which is invertible in $R$ is invertible ...
2
votes
0answers
157 views

What is it called when a subalgebra contains its centralizer?

In the question Math.SE #16716, Natalia asked about representing rings of matrices as centralizers of a matrix. This is an intriguing question, but had some clear problems as rings of matrices need ...
4
votes
1answer
377 views

How do you show the ring of formal laurent series is well-defined?

The only place I've encountered well-definition is with proving an operation defined on an equivalence class is independent of the choice of representative. On my homework, it asks us to show that ...
14
votes
3answers
930 views

Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number?

I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\{0\}$ ...