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

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4
votes
4answers
125 views

Why are roots of polynomials called geometric objects?

I read the following from the Wikipedia article about algebraic varieties: Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by ...
0
votes
3answers
48 views

Easier way to show $(\mathbb{Z}/(n))[x]$ and $\mathbb{Z}[x] / (n)$ are isomorphic

$$(\mathbb{Z}/(n))[x] \simeq \mathbb{Z}[x] / (n)$$ I've shown this by showing that the map that sends $\overline{1} \mapsto [1+(n)]$ (where the bar denotes the congruence class mod $n$) and $x ...
1
vote
1answer
60 views

Quotient of $\mathbb{C}[x]$ by $(x-a)^2$ or by $x^2$.

Are $\mathbb{C}[x]/((x-a)^2)$ and $\mathbb{C}[x]/(x^2)$ isomorphic? I see that the elements (cosets) of each rings can be identified with linear polynomials $c_1 x + c_0$, but I am not sure what to ...
3
votes
1answer
116 views

Smallest skew-field containing a non-commutative ring.

Let $R$ be an integral domain and take $D = R - \left\{ 0 \right\}$. The ring $D^{-1}R$ is the smallest field containing $R$ as a subring. Now suppose that I have a non-commutative ring $N$. Suppose ...
2
votes
0answers
269 views

What is the fraction field of formal power series ring over a field $F$?

The field of fractions of the formal power series ring $F[[x]]$ over a field $F$ can be obtained by inverting the elements $x$. Let $R=F[[x]]$. I have difficulty in finding the isomorphism $$ ...
0
votes
1answer
72 views

Multiplicative identity of a quotient ring

Let $R$ be a commutative ring such that $M$ is a maximal ideal of $R$. Then I know that $R/M$ is a field. But I am unable to understand what is the multiplicative identity of $R/M$? If $R$ has an ...
2
votes
1answer
165 views

Maximal ideals of rings which are finite dimensional vector spaces over $\mathbb C$.

If $K$ is a commutative ring which is a finite dimensional vector space over $\mathbb C$ what can we say about the maximal ideals of $K$? What can we say if instead of $\mathbb C$ we have some ...
2
votes
2answers
100 views

What is the structure of matrix multiplication and minus?

Please note I have only little background im mathematics and I am working on formalizing theorems with theorem provers. This is very much a beginner question. Suppose I have matrices, where the ...
3
votes
2answers
74 views

Is this true: $\mathbb{Q}$[$\sqrt[4]{5}$(1-$i$)] = $\mathbb{Q}$[$\sqrt[4]{5}$(1+$i$)]

$\mathbb{Q}$[$\sqrt[4]{5}$(1-$i$)] = $\mathbb{Q}$[$\sqrt[4]{5}$(1+$i$)] ? Having no clue how to proceed. Obviously shouldn't be using plain algebra to expand the terms? Thanks!
2
votes
1answer
100 views

Does the natural bijection between the set of prime ideals in A disjoint from S and Spec$(S^{-1}A)$ restrict to maximals?

I was studying rings of fractions, and I was wondering about the problem of restricting the canonical bijection (induced by retraction and extension of ideals) $\{p\in \text{Spec}(A) \mid p\cap ...
2
votes
1answer
178 views

Left/right inverses in non-commutative ring

Let the setting be a non-commutative unital ring $A$ and $a,b \in A$. Assume $a$ is invertible, meaning it has either a left inverse, a right inverse or both. Does it then hold that $ab$ is ...
2
votes
1answer
26 views

Nonexistence of the lcm(p, px)

Let $p \in \mathbb{ Z }$ be a prime number. Define $R \subseteq \mathbb{ Z }[X]$ to be the integral domain such that for any element the coefficient of $x$ is divisible by $p$. I'm supposed to show ...
2
votes
1answer
58 views

Adjoin to a ring an element satisfying a monic vs. non-monic polynomial.

Artin defines the adjoining of an element $\alpha$ to ring $R$ satisfying polynomial $f \in R[x]$ by $$R[\alpha] = R[x]/(f).$$ If $f$ is monic with degree $n$, then we get some nice properties, such ...
1
vote
0answers
30 views

representation type of PI rings

A ring $R$ is said to satisfy a polynomial identity (PI for short) if there exists a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z} \langle X_1, \ldots, X_n \rangle$ such that $f(r_1, \ldots, r_n)=0$ ...
2
votes
2answers
116 views

Integral dependence over rings is transitive

Let $A\subset B\subset C$ be commutative rings. Suppose $B$ is integral over $A$, and $C$ is integral over $B$. Then I want to show that $C$ is integral over $A$. To be integral means that for ...
0
votes
3answers
68 views

Ideals of $R/N$

I have a proof in a book I'm reading that says (we are dealing with ring $R$ and ideal $N$) that the ideals of $R/N$ are of the form $M/N$ where $M$ is an ideal of $R$ that contains $N$. Can someone ...
0
votes
3answers
90 views

Non-zero prime ideals of $F[x]$ are maximal

Prove that if $F$ is a field, every proper prime ideal of $F[X]$ is maximal. Should I be using the theorem that says an ideal $M$ of a commutative ring $R$ is maximal iff $R/M$ is a field? Any ...
0
votes
1answer
88 views

How to show $1 + \sqrt{ 5 }$ is irreducible

I'm trying to show that $p = 1 + \sqrt{ 5 }$ is irreducible in $\mathbb{ Z }\left(\sqrt{ 5 }\right)$. After applying the norm $N\left(a + b\sqrt{ 5 }\right) = a^2 - 5b^2$, I find that $N(p) = -4 = -2 ...
1
vote
2answers
138 views

About proving that maximal ideals are prime

Let $R$ be a Ring with unity, and $I$ a maximal ideal in $R$. Show that $I$ is a prime ideal. I have seen this proof in many places If $ab\in I$ and $a\notin I$, then $I+(a)=R$ and hence there is ...
3
votes
2answers
210 views

Ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$

What is the ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$? So, these are numbers of the form $a+b\sqrt{3}+c\sqrt{23}+d\sqrt{69}$ where $a,b,c,d\in\mathbb{Q}$, and we want to find ones whose ...
0
votes
1answer
35 views

Ideal of a field

Let $F$ be a field. Show that $S$ be a non empty subset of $F^{n} $ then $ I(S) =$ { $ f(x) \in F[x] \hspace{0.1in} \vert \hspace{0.1in}f(s) = 0 \hspace{0.1in} \forall s \in S $ } is an ...
0
votes
1answer
118 views

Relation between Jacobson radical and composition series

Let $R$ be a not necessarily commutative ring with 1. Suppose $R$, viewed as a right $R$-module, has a finite composition series with non-isomorphic composition factors. Prove that the Jacobson ...
1
vote
1answer
47 views

Semisimples rings and units

Is possible to proof directly of definition of semiplicity this: if R is a semisimple ring, then R have unit?
6
votes
3answers
292 views

Finding inverse in non-commutative ring

Let $a,b,c,x$ be elements of a unital non-commutative ring. Assume $c$ is an inverse of $1-ab$: $$ c(1-ab) = 1$$ How can I find an inverse for $1-ba$? What I tried: Denote the unknown by $x$. Then ...
11
votes
1answer
141 views

$B\otimes_A A[x]=B[x]$

Let $A\rightarrow B$ be a homomorphism of commutative rings. Then $B\otimes_A A[x]\cong B[x]$ as $B$-algebras. How can one demonstrate this nicely, i.e. using universal properties alone and the Yoneda ...
6
votes
2answers
894 views

Prime elements in $\mathbb{Z}[\sqrt{2}]$

What are the prime elements in the ring $\mathbb{Z}[\sqrt{2}]$? Note that since the ring is a PID (and thus a UFD) then prime = irreducible. Even more, it is Euclidean with respect to the absolute ...
0
votes
1answer
36 views

Adjunction: Why does modding out by a polynomial “add” the zero of the polynomial. [duplicate]

Suppose we have a ring R and we form a polynomial ring in R, R[x]. In class my professor has explained that forming the quotient ring of R with some polynomial f(x), is like adding the zero of that ...
2
votes
1answer
57 views

Roots of polynomials in field of prime characteristic p

Let $F$ be a field of characteristic $p$, and let $\alpha \in F$. Let $f \in \mathbb{Z}_p[x]$ be such that $f(\alpha) = 0$. Apparently, we have $f(\alpha^p) = 0$. This was mentioned but not proved in ...
0
votes
1answer
71 views

Determining if ___ is an ideal:

I am reading some notes on algebra, with polynomial rings and there is an exercise that asks to determine if 1-5 are ideals. I am not very familiar and am hoping that the examples will give a more ...
3
votes
1answer
83 views

Ring of Polynomials Commutative?

Can $k[x_1,...,x_n]$, the ring of polynomials with coefficients $\in k$ where $k$ is a field, ever be a non-commutative ring?
3
votes
0answers
53 views

$G \simeq R^{\times}$

What is known about the groups G for wich there exist a unitary ring R, such that $R^{\times} \simeq G$? I can easily prove that The only G cyclic with this property(Edit:and odd order) are those who ...
3
votes
1answer
116 views

Cubic field and the corresponding cubic binary form

I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...
4
votes
1answer
112 views

Maximal ideals of the ring of formal polynomials over a ring $R$

Consider the polynomial ring $R[x]$ over a ring $R$. Then $I$ is a maximal ideal of $R[x]$ if and only if $R[x]/I$ is a field. When $F$ is a field, we know that $I$ is of the form $(f)$ where $f(x)$ ...
2
votes
1answer
62 views

A question on rings

Let $R$ be an integral domain and $S$ be subring of $R$ with $1_R=1_S.$ Let $T=\{f(x) \in R[X]: f(0) \in S\}.$ Suppose $R[X]$ satisfies ascending chain condition for principal ideals, $ACCP.$ Could ...
2
votes
1answer
82 views

gcd's in non-UFD rings

In a UFD ring we have that for coprime $a,b \in R$, i.e. $(a,b)=1$: $$ a|cb \Rightarrow a|c $$ Does this property hold for non-UFD rings? I think not but do not recall a standard ...
7
votes
2answers
114 views

If $k\subset R\subset k[x]$, then $R$ is Noetherian?

Is there a way to prove that any subring $R$ of the polynomial over a field $k$ such that $k\subset R$ is Notherian without appealing to integral extensions, Eakin-Nagata, etc.? The reason I ask is ...
0
votes
2answers
45 views

Matrix ring over a ring

Let $k$ a ring with $1\neq{0}$. Wich is the center of $M_n(k)$? I'm suppose that $Z(M_n(k))=\{r\cdot{Id} \ : \ r\in{Z(k)} \}$ Is correct? Thanks.
1
vote
1answer
37 views

Proper factors and subsets of integral domains

We want to prove that if $R$ is an integral domain (with identity element $1_R$), then $a$ is a proper factor of $b$ (a proper factor meaning, there exists $c$ in $R$ such that $b = ac$, and $c$ is ...
10
votes
2answers
124 views

Real forms of complex vector spaces and $\mathbb{C}$-algebra

A real form $W$ of a complex vector space $V$ is a real subspace s.t. $\mathbb{C}\otimes_{\mathbb{R}}W \cong V$ by $a\otimes x \longrightarrow ax$, or equivalently there is an $\mathbb{R}$-basis of ...
6
votes
1answer
153 views

The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring ...
2
votes
1answer
181 views

Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $ g := x^n - 1 \in F[x]$ Is it true that $ g$ has distinct roots in $F$ if and only if ...
1
vote
0answers
64 views

Faithfully flat checkable on finitely generated modules

A left $R$-module $_RM$ is said to be faithfully flat if it is flat and, for any $N_R$, $N \otimes_R M = 0$ implies $N = 0$. I would like to show that $M$ is faithfully flat if it is flat and, for any ...
1
vote
1answer
125 views

Quotient rings in formal power series

I'm trying to find all prime ideals in the formal power series ring $k[[x]]$, where $k$ is a field. I think I've managed to show that all ideals are of the form $<x^n>$, $n>0$, i.e. ...
1
vote
0answers
64 views

dimension of tensor products over a submodule

Let $k$ be a field, $A$ be a finite dimensional $k$-algebra (say of dimension $n_A$) and let $B\subset A$ be a sub-algebra. What can be said about the $k$-dimension of $A \otimes_B A$ ? The easiest ...
1
vote
0answers
46 views

Enveloping Algebra equal to algebra

Let $R$ be a unital associative ring, $A$ be an associative $R$-algebra of finite dimension, and $A^e$ its enveloping algebra. What are the requirements on $A$, so that $A^e \cong A$ (as ...
1
vote
1answer
176 views

How to construct the subring generated by a set, T?

I'm trying to find a constructive way of describing the subring generated by some subset, T, of a ring R. I think I could describe it as all finite sums of finite products of elements of T, but I ...
0
votes
1answer
53 views

Consider the ring $R=\mathbb{Q}[x]$

$\newcommand{\QQ}{\mathbb{Q}}$ Consider the ring $R=\QQ[x]$ over $I$ where $I$ is an ideal of $\QQ$ generated by the polynomial $x^2-2$ in $\QQ[x]$. (a) Show that for all $f(x)$ in $\QQ[x]$, ...
1
vote
1answer
53 views

I as an ideal of $R$ then $a+I=0+I$ iff $a\in I$ [duplicate]

show that if a,b belong to the ring $R$ and $I$ is an ideal of $R$ then $a+I=0+I$ if and only if $a$ belongs to $I$. I know that since I is an ideal then it is both a left and a right ideal.
2
votes
4answers
90 views

In the proof of “maximal ideals of $\Bbb{Z}[x]$ are of the form $(p,f(x))$ ”

I'm trying to prove the following statement: Maximal ideals of $R=\Bbb{Z}[x]$ are of the form $(p,f(x))$ where $f(x)$ is irreducible in $\Bbb{F}_p[x]$ and $p$ is prime. A quick search on google ...
3
votes
3answers
287 views

Nontrivial ideal of a Noetherian domain contains a finite product of nonzero prime ideals

If $R$ is a Noetherian domain and $ 0 < U < R$ an nontrivial ideal of $R$. How to prove that there exists nonzero prime ideals $p_1,...,p_n \subset R$ such that the product $ p_1 p_2 ...p_n ...