# 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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### 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 ...
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### 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?
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### “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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $P_P=S^{-1}P$, ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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.
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### 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 ...