Questions tagged [ring-theory]

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

A ring $R$ is a triple $(R,+,\cdot)$ where $R$ is a nonempty set such that $(R,+)$ forms an abelian group, $(R,\cdot)$ forms a semigroup, and the two operations are related by the distributive laws: $a\cdot(b+c)=a\cdot b+a\cdot c$ and $(b+c)\cdot a=b\cdot a+c\cdot a$.

Important examples of rings include domains (such as the integers), fields (such as the real numbers), square matrix rings, polynomial rings, and rings of functions. Rings are studied in their own right in abstract algebra, but they are also prominently used in number theory, geometry, algebraic geometry, and logic.

Many authors require the semigroup $(R,\cdot)$ to have an identity, often denoted $1_R$ or $1$. Many other authors do not make that requirement. This difference is something that students and posters should be aware of. Scholars of the former school call the structures not necessarily having a unit element . Scholars of the latter school call $R$ a ring with identity, when $1_R$ exists. This difference of opinions has an impact on the definition of a ring homomorphism. The scholars who include the presence of $1_R$ as an axiom assume that it is preserved under ring homomorphisms. The scholars who don't insist on the existence of $1_R$ obviously cannot make this requirement.

The operation $\cdot$ does not have to be commutative, but when it is, $R$ is called a commutative ring.

There are numerous types of rings studied in different ways. An ideal in a ring is the ring-theoretic analogue of a normal subgroup of a group. The study of ideals is an important component of ring theory.

This tag often goes along with the and/or tags.

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Ideal in a Noetherian ring which is not equal to a product of prime ideals

In a Noetherian ring, it is known that every ideal contains a product of prime ideals. Is there any example of a Noetherian ring in which an ideal is not equal to any product of prime ideals? This is a natural question,which I didn't find even as…
Beginner
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Showing that a number is prime if norm is prime

Suppose that $\alpha\in\mathbb{Z}[i]$, and $N(\alpha)=\alpha .\bar{\alpha} =p$ (norm), a prime in $\mathbb{Z}$. Then show that $\alpha$ is a prime in $\mathbb{Z}[i]$. My attempt: Let $\alpha|ab$ for some $a,b\in\mathbb{Z}[i]$ and if we assume…
user428700
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A finitely-generated ring over the integers modulo a max ideal is necessarily a finite field

Suppose that you have a ring $\Lambda \subset \mathbf{C}$ such that $\Lambda$ is finitely generated over $\mathbf{Z}$. Let $\mathfrak{m}$ be a maximal ideal of $\Lambda$. Then the field $\Lambda / \mathfrak{m}$ must be finite. Serre uses this…
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Is $\mathbb{Z}[\sqrt{29}] $ a PID

Question as in title. I think that unique factorization fails, perhaps via either $ (\sqrt{29} - 1)(\sqrt{29} + 1) = 2^2 \cdot 7 $ or $ (\sqrt{29} - 5)(\sqrt{29} + 5) = 2^2 $, but I have trouble proving either of these two claims. How to solve this…
user228960
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Examples of interesting rings to study during an undergraduate course in non-commutative rings.

I'm taking a course on Modern Algebra at my university and my professor keeps asking us to look for examples of rings that may be interesting to discuss in class. For instance, he dedicated some time today to discuss upper triangular rings which has…
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Is the set of all Irrational Numbers a ring or a field?

I would really appreciate a proof of either one. I think it should be a field as it satisfies the multiplicative and additive identities and is commutative.
BehavingEarth
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Prove $u$ is a unit if and only if $N(u) = 1$?

The norm $N:\mathbb{Z}[\sqrt[3]{2}] \rightarrow \mathbb{N}$ defined by $N(a+b\sqrt[3]{2} + c\sqrt[3]{4}) = |a^3 + 2b^3 + 4c^3 - 6abc|$ is multiplicative. (Already proven). Show that $\alpha \in \mathbb{Z}[\sqrt[3]{2}]$ is a unit if and only if…
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Showing $\mathbb{Z}[i]/(1+2i) \oplus\mathbb{Z}[i]/(6-i)\cong\mathbb{Z}[i]/(8+11i)$

I am attempting to solve Ch 14 Problem 7.7 from Artin's algebra book. Let $R=\mathbb{Z}[i]$ and let $V$ be the R-module generated by elements $v_1$ and $v_2$ with relations $(1+i)v_1+(2-i)v_2=0$ and $3v_1+5iv_2=0$. Write this module as a direct sum…
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Is $\bigcap_{n=1}^\infty I^n$ idempotent, for any ideal $I$?

Let $I$ be an ideal in a ring $R$. By $I^n$, let us understand the ideal generated by all $n$-fold products $x_1x_2\cdots x_n$ where $x_1, \ldots, x_n \in I$. Obviously $I \supseteq I^2 \supseteq I^3 \supseteq \ldots$. Let us also define $I^\infty…
Mike F
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Zero divisors and minimal prime ideals in commutative ring

Let $A$ be a commutative ring with unity different from $0$. Let $D(A)$ be the set of those prime ideals $\mathfrak{p}$ of $A$ which satisfy $$(*) \mbox{ there is } a\in A \mbox{ s.t }. \mathfrak{p} \mbox{ is a minimal among prime ideals…
Beginner
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$p(x)x^n+q(x)(1-x)^n=1$ for some $p(x), q(x)\in \mathbb{Z}[x]$, what explicitly $p(x), q(x)$ are?

Because $x^n$ and $(x-1)^n$ are relatively prime in $\mathbb{Z}[x]$, so $p(x)x^n+q(x)(1-x)^n=1$ for some $p(x), q(x)\in \mathbb{Z}[x]$. What are the explicit formulas of $p(x), q(x)$?
HeHe
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In what algebraic structure does repeated addition equal multiplication?

I'm trying to figure out for which algebraic structure $$\underbrace{a+a+\cdots+a}_{n \text{-times}} = a * n$$ is true. Now I know the question 'Is all multiplication repeated addition?' has been asked many times with the answer: NO because you…
Ryan Stull
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Proof that ring is commutative

Proof that a ring $(A, + , \cdot)$ is commutative if $$x(y^2+y)=(y^2+y)x$$ I set $x$ to be $$(x-xy^2) $$and I get that $$xy^3=yxy^2.$$But I don't know how to get $xy=yx$.
nnMan
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Structure theorem for modules implies Smith Normal Form

I am working on a homework problem for an undergrad abstract algebra course and I have been stuck on something for a while. I am not looking for a full proof, but any help would be much appreciated. Apologies if this is something simple! The…
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Identify $\mathbb{Z}[x]/(2x^2+1,2x-3)$.

Identify $\mathbb{Z}[x]/(2x^2+1,2x-3)$. I tried this: $$ \mathbb{Z}[x]/(2x^2+1,2x-3)=:R \\ (2x^2+1,2x-3)=:I \\ \because (2x^2+1)-x(2x-3)=3x+1, \\ 2(3x+1)-3(2x-3)=11 \\ \therefore I=(2x^2+1,2x-3,11) \\ \therefore R \cong…
sfz
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