2
votes
1answer
35 views

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) ...
23
votes
6answers
1k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
0
votes
1answer
25 views

Definition of a ring S being finitely generated as an R-algebra

What is the definition of a ring S being finitely generated as an R-algebra, where R is a ring?
1
vote
0answers
48 views

“Filters” of associative rings

Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings? For me, if $R$ is an associative ring with unity $1$, it ...
0
votes
0answers
26 views

What is the definition of a norm in the context of rings?

On several places in ring-theory I encountered so-called 'norms'. For instance on integral domain $\mathbb{Z}\left[i\right]$ with prescription $a+bi\mapsto a^{2}+b^{2}$ where it also serves as a ...
0
votes
0answers
43 views

Modules,rings and definitions

Is there a source available with (almost) all definitions from ring and module theory,all in ONE place without theorems.There are books on module theory where are freely used unusual notions like ...
0
votes
1answer
67 views

Why Integral domains haven't an unified definition? [duplicate]

We can define integral domains as: rings without zero divisors commutative rings without zero divisors commutative rings with identity and without zero divisors I don't know why integral domains ...
0
votes
2answers
42 views

Definition of localization of rings

I'm trying to understand this definition of Hungerford's book: The definition is simple, I think I understood what the author means, but... What is $P_P$? because we will have $P_P=S^{-1}P$, ...
2
votes
1answer
140 views

What is the exact definition of polynomial functions?

I'm trying to understand the difference between polynomial functions and the evaluation homomorphisms. I noticed that I don't know what's the exact definition of a polynomial function, although I've ...
2
votes
2answers
97 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
95 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 ...
1
vote
1answer
43 views

If a ring $R$ is a field, must $R$ be a unitary ring?

If a ring $R$ is a field, then does it automatically imply that $R$ is a unitary ring? Thank you.
4
votes
2answers
59 views

Question on the definition of a ring.

A ring $\langle R,+, \cdot\rangle $ is a set $R$ with two binary operations such that: $\langle R,+\rangle $ is an abelian group. Multiplication is associative. Left and right distributive laws ...
8
votes
4answers
414 views

A question on definition of field of fractions

Wikipedia defines the field of fractions of a domain as The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded. What does ...
14
votes
2answers
3k 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, ...
3
votes
2answers
179 views

Understanding of extension fields with Kronecker's thorem

In the book Contemporary Abstract Algebra by Gallian it defines an extension field as follows: A field $E$ is an extension field of a field $F$ if $F\subseteq E$ and the operations of $F$ are ...
2
votes
3answers
151 views

Is the zero of a commutative ring not a zero divisor or is it “undefined?”

In the Contemporary Abstract Algebra book by Gallian it defines zero-divisors as follows: Definition 1) A zero-divisor is a nonzero element $a$ of a commutative ring $R$ such that there is a ...
2
votes
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 ...
0
votes
2answers
95 views

Definition clarification on ideals

Suppose $P$ is the set of all subsets of a set $X$ and $P$ is a ring. Let $p$ be an element in $P$ (so that $p$ is a subset of $X$). What does it mean to say "an ideal generated by $p$"? And suppose ...
6
votes
5answers
551 views

Definition of Ring

I'm studying Abstract Algebra right now, currently covering rings. In the introduction of the subject, I am curious as to why there is no need for there to be a multiplicative identity. I understand ...
15
votes
3answers
568 views

Generalization of a ring?

I've just started learning about rings. Rings are one additive abelian group strung together, through the associative law, with another structured operation. Couldn't we continue stringing together ...
17
votes
3answers
1k 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, $0\cdot\mathbb Z=\{0\}$ ...