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.
21,313
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If any ideal in $R$ is principal, then any ideal in $R \times R$ is principal
Let $R$ be a ring. We endow $R \times R$ with pointwise addition and pointwise multiplication. Then it's easy to verify that $R \times R$ is a ring under these operations. Then I conjecture that
If ...
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Is commutativity needed in the proof of division algorithm?
I found that my 2 textbooks and other answers state this theorem for commutative ring:
From textbook Algebra by Saunders MacLane and Garrett Birkhoff.
From textbook Analysis 1 by Herbert Amann and ...
5
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1
answer
267
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Algebra of upper triangular matrices
Let $k$ be a field and $R$ the algebra of $3\times3$ upper triangular matrices $(a_{ij})$ st $a_{11}=a_{22}=a_{33}$.
Find the Jacobson radical $J(R)$ of $R$
Attempt: Using the characterization $y\in ...
4
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2
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311
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Does the unit generate the additive group in a unital ring with cyclic additive group?
Let $R$ be a unital ring with cyclic additve group $(R, +,0)$. Is it the case that $1$ generates the additive group $(R,+,0)$?
Thoughts:
Maybe classifying the unital rings with cyclic subgroups is ...
1
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506
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Are the ideals of a ring with cyclic additive group always principal?
Note for me rings need not be unital or commutative.
Let $R$ be a ring with cyclic additive group $(R, +, 0)$ and let $I$ be an ideal in $R$. Is $I$ principal?
Here's my attempt, assuming $R$ has a $...
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2
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100
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How to find a ring with a given automorphism group
I'm hoping to find an example of a ring whose automorphism group is D8, i.e. the order 8 dihedral group. This raises the question of how, given a group $G$ to find a ring $R$ whose automorphism group ...
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In a finite commutative ring , every prime ideal is maximal?
I am stuck in a true/false question. It is
In a finite commutative ring, every prime ideal is maximal.
The answer says it's false.
Well what I can say is (Supposing the answer is right)
$(1)$ The ring ...
2
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1
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If the associated graded algebra of a filtered algebra is commutative, is the condition below satisfied by the filtration?
Suppose we're given a filtered algebra $A$ over a field $k$ with filtration $F_{\bullet}A$ over the subspaces of $A$: $$\{0\}\subseteq F_{0}A\subseteq\cdots\subseteq F_{i}A\subseteq \cdots\subseteq A,$...
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Show that any endomorphism of $\mathbb Q[x]$ is $E_{g}$ for some $g$, and find all automorphisms of $\mathbb Q[x]$
I'm doing Exercise 3 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
For a commutative ring $R$ and $c \in R[x]$, the map $E_c: R[x] \to R[x]$ is a homomorphism such that $E_c(r) = r$ ...
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Proving that central simple algebra $B$ over a field $k$ splits in Brauer group by a field $L$ contained in $B$, then $[B:k]=[L:k]^2$
I want to prove the following statement (I read it here - https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.4.3.6)
The similarity in the below statement comes from the equivalence relation from the ...
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The algebra $Z(A)/J(Z(A))$ is a product of field extensions of $k$
Let $A$ be a finite-dimensional $k$-algebra. If $A$ is split, then $Z(A)$ is split.(Let $A$ be a finite-dimensional $k$-algebra. We say that $A$ is split if $\operatorname{End}_A(S) \cong k$ for every ...
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$M_n(k)$ is an indecomposable algebra
Let $A$ be a $k$-algebra. The following are equivalent:
(i) $A$ is indecomposable as a $k$-algebra.
(ii) $A$ is indecomposable as an $A-A$-bimodule.
(iii) The idempotent $1_A$ is primitive in $Z(A)$.
...
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$K$ is not endowed with any algebraic structure. How can pointwise sum and convolution product be defined on $K^{(S)}$
I'm doing Exercise 6 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
If $S$ is an additive monoid, the support of a function $f: S \rightarrow K$ is the set of all those $s \in S$ with $...
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When does a dominant morphism of varieties induce a free extension of coordinate algebras?
Let $\phi : X \to Y$ be a dominant morphism of irreducible varieties over an algebraically closed field $\mathsf{k}$, and for the sake of simplicity, let us assume that $X$ and $Y$ are affine. Then ...
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196
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Zero is always a Superfluous submodule.
Definition:-1 A submodule $N$ of a module $M$ is said to be superfluous (or small) if there is no proper submodule $K$ of $M$ such that $M=N+K$.
Definition:-2 Jacobson radical $J(M)$ of a module $...
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2
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111
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Nontrivial subring of $\mathbb{R}$ not containing $1$
Are there examples of nontrivial subrings of $\mathbb{R}$ that do not contain $1$? If not, how can we prove they don't exist? The definition of "ring" here is really "rng"; rings ...
1
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27
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Almost principal ideals
Suppose $R$ is a non-unital ring. Then given an $r\in R$, we may have $r\notin Rr$, so $Rr$ is not necessarily the left ideal generated by $r$.
It is not hard to see that $Rr$ may not even be a ...
2
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2
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144
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What is the formal way of going from $(b_0, b_0+b_1p, b_0+b_1p+b_2p^2, \dots) \in \mathbb Z_p$ to the formal sum $b_0 + b_1 p + b_2 p^2 + \cdots$?
I've figured out how to write an element of the $p$-adic integers $(a_1, a_2, a_3, \dots) \in \mathbb Z_p=\varprojlim \mathbb Z/p^i\mathbb Z$ as
$$(b_0, b_0+b_1p, b_0+b_1p+b_2p^2, \dots)$$ by ...
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Definition of algebra of polynomial functions on a vector space.
In his lectures nots on Topology, Prof. Nacinovich gives as an example of topological space the Zariski Topology on a vector space. To this end, he says
Let $V$ a finite dimensional vector space over ...
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Show that there is a unique morphism $E_{c}: R[x] \rightarrow R^{\prime}$ with $E_{c}(r)=r$ for all $r \in R$ and $E_{c}(x)=c$
I'm doing Exercise 5 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
If a ring $R$ is not commutative, an element $c$ is called central in $R$ if $c r=r c$ for every $r \in R$. ...
2
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1
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196
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Subrings of a Noetherian ring which inherits the Noetherian property
The following is exercise $1.5$ in the book Commutative Algebra With A View Toward Algebraic Geometry by David Eisenbud.
Exercise 1.5
Let $S$ be a commutative Noetherian ring and $R\subset S$ be a ...
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1
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147
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algebraically dependent and transcendental
Let $T$ be transcendental basis of $R$ over $\mathbb Q$ and $A\subset T.$ Consider $$S=\{a^{-1}\colon a\in A\}$$ 1 Is S algebraically independent over $\mathbb Q$ ?
My second question: If $B\subset\...
2
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3
answers
234
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Lang's definition of generated subring (?)
Let $A$ be a subring of a ring $B$. Let $S$ be a subset of $B$
commuting with $A$; in other words we have $as=sa$ for all $a\in A$
and $s\in S$. We denote by $A[S]$ the set of all elements $$\sum
a_{...
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650
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Multiple root of a polynomial and formal derivative.
I've come across this theorem in Hungerford's Algebra textbook.
Let $D$ be a field which is a subring of an integral domain $E$. Let $f \in D[x]$ ($D[x]$ is the polynomial ring). If $f$ is ...
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2
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57
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we may assume that $J(A) = 0$
$k$ is a field with at least 3 elements. $A$ finite-dimensional $k$-algebra. Then $A$ has a $k$-basis consisting of invertible elements of $A$.
In the proof, the author shows first that the propertiy ...
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Examples of non-commutative unity rings where $2$ is a non-zero zero divisor
I would like to see some examples of non-commutative unity rings where $2$ is a non-zero zero divisor.
My thoughts on this: Since $2$ is a zero divisor there will be some $x$ in that ring such that $...
1
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1
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73
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$A/J(A)$ is semisimple as an $A$-module, then $A$ is semisimple
$A/J(A)$ is semisimple as an $A/J(A)$- or $A$-module, then $A$ is semisimple.
Is this even true?(by the counterexample kindly provided below this is not.) I am asking this because I've a corollay of ...
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295
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Understanding surjectivity in Chinese Remainder Theorem Proof
Im self-studying some ring theory and in Dummit and Foote Chapter 7.6 Theorem 17, pg 265-266 they have a proof for the case where $k = 2$ that I am struggling to understand--maybe I'm forgetting some ...
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Since each $M_n(D)$-submodule of $M_n(D)$ is also a $D$-submodule, we have $M_n(D)$ artinian as a ring.
$D$ is a division ring so it is artinian, hence so is $M_n(D)$ as a $D$-module. Since each $M_n(D)$-submodule of $M_n(D)$ is also a $D$-submodule, we have $M_n(D)$ artinian as a ring.
I am not sure ...
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Finding a group isomorphic to ring of polynomials
let $J$ be an ideal of $F[x]$ such that $J$ contains every polynomial of $\mathbb{Z}[x]$ with constant term a multiple of 4. Im looking for a ring isomorphic to $\mathbb{Z}[x]/J$, with proof.
I was ...
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Question about multiplication of elements in the associated graded ring and associated graded module.
If $F_{\bullet}R$ is a filtration of a ring $R$, the associated graded ring of $R$ is defined as $$
\mathrm{gr}_{\bullet}(R):=\bigoplus_{i \in \mathbb{N}_{0}} \mathrm{gr}_{i}(R),
$$
where $\mathrm{gr}...
3
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1
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Weil Restriction and Distinguished Opens
I have a pair of related questions about Weil Restriction. Let $E/F$ be a field extension, and let $A$ be an $E-$algebra. Assume that all relevant restrictions of scalars exist. We have a norm map $n: ...
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0
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67
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Is there a non-commutative discretely ordered ring?
An ordered ring is a (not necessarily commutative) ring with a total order $≤$ respecting the operations:
if $a ≤ b$ then $a + c ≤ b + c$,
if $0 ≤ a$ and $0 ≤ b$ then $0 ≤ ab$ (equivalently: if $a ≤ ...
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Need help understanding the structure of this finitely generated graded algebra.
Suppose $R$ is a ring with filtration $F_{\bullet}R$: $$
\{0\} \subseteq F_0R \subseteq F_1R \subseteq \cdots \subseteq F_{n}R \subseteq \cdots \subseteq R.
$$
Let $\mathrm{gr}_{\bullet}^{F}R$ be the ...
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Does there exist a field ${K[X]}/{(p)}$ where $p$ is reducible?
Suppose a ring $K$. Does there exist a field ${K[X]}/{(p)}$ where $p \in K[X]$ and p is reducible?
I know that as soon as $(p)$ is a maximum ideal from $K[X]$ that ${K[X]}/{(p)}$ is a field. ...
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2
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Is this version of evaluation homomorphism theorem correct?
The theorem comes from Kostrikin's Introduction to Algebra and goes as follows:
If $A$ (a commutative ring with an identity) is a subring of a commutative ring $R$, then for every element $t \in R$ ...
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0
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Is the ideal $R= (x+y, x−y)$ in the polynomial ring $\mathbb{C}[x, y]$ a prime ideal?
I wanted to check if my approach is correct:
Consider : $$\phi:\mathbb{C}[x, y] \rightarrow \mathbb{C}[x]$$ $$ y \mapsto -x$$ $$x \mapsto x$$ is a surjective homomorphism, with $\ker(\phi)=(x+y)$. ...
1
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1
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78
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Is simple submodules of a discrete module a direct summand of the module?
Definition::An $R$ module $M$ is said to be discrete if it satisfy following two properties
$(D_1)$ For every submodule $A$ of $M$, there is a decomposition $M=M_1\oplus M_2$ such that $M_1\leq A$ and ...
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Is the following ring a prime/maximal ideal?
Are the following statements true or false in rings $R \subset S$ with unity $1$?
If $I\vartriangleleft R$ is a prime ideal, then $\{ \sum_{i \in \mathbb N}r_is_i| r_i \in I, s_i \in S\}$ is a prime ...
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0
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254
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Quotients of Two-sided and One-sided Ideals
Let $\mathcal{A}$ be a central simple algebra over an algebraic number field $K$, and $\mathcal{O}$ be a maximal $\mathcal{O}_K$-order in $\mathcal{A}$. Let $I$ be a maximal integral left-ideal of $\...
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1
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43
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Formal powers series in $n$ indeterminates
Let $R\ne\{0\}$ be a ring with identity and $2\leq n$ an integer.
For each $1\leq i\leq n$, let $\lambda_i:[1,n]\rightarrow R$ be the mapping for which $\lambda_i(i)=1$ and $\lambda_i(j)=0$ for $j\ne ...
2
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2
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239
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Find all integers such that polynomial $x^4+n$ is reducible in $\mathbb{Z}[x]$
In my studies of abstract algebra and polynomials, I have the following question:
We are asked to find all integers $n$ such that the polynomial $x^4+n$ is reducible in $\mathbb{Z}[x]$.
The only ...
1
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1
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68
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$\mathbb{Z}_{3}[T]$/$(T^{2}+2T+2)$ is a commutative field
I'm trying to prove that $K$= $\frac{\mathbb{Z}_{3}[T]}{(T^{2}+2T+2)}$ is a commutative field.
So I thought i needed to prove that $K$ is commutative and $K$ is a ring with unit element so that for ...
0
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1
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32
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$X$ is finitely generated as a module over $A$.
$A,B$ are $k$-algebras and finitely generated as $k$-modules. Let $X$ be a simple $A \otimes_k B$ module. Since $A \otimes_k B$ is finitely generated as a $k$-module, $X$ is finitely generated as a $k$...
1
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1
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59
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Computation in a Quotient Ring
I have $f(x)=x^3 +x -2 \in \mathbb{Q[x]}$ and R is the ring $\mathbb{Q[x]}/\langle f \rangle$.
$\alpha = x +\langle f \rangle$ is the image of $x$ in R. Note R is not a field as $f(x)=(x-1)(x^2 +x+2)$....
0
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1
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256
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Proving that a ring homomorphism $R[X] \to R^R, p \mapsto \underline p$ takes $1$ to $1$
This has to do with Remark 8.14 (b) on page 75 of Analysis I by Amann and Escher.
I am not defining all the notation used because I'm trying to keep the length of this post manageable.
Excerpt from ...
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1
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How can I prove that if $M$ is a graded $R$-module, then each $M_i$ is also finitely generated?
I want to prove the following:
If $M=\bigoplus_{i \in \mathbb{Z}} M_i$ is a finitely generated graded
$R$-module ($R=\bigoplus_{i \in \mathbb{N}_0} R_i$ is also a graded
ring and finitely generated ...
2
votes
1
answer
224
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Understanding a variant on the multinomial theorem in a commutative ring with unity
This post concerns Chapter 1 section "The Multinomial Theorem" on pages 65-67 of Analysis I by Amann and Escher.
Excerpts from text:
The part that I can't understand is the equation with the ...
1
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1
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468
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Proving operations of a quotient ring are well-defined
I'm a little confused about a specific bit of reasoning in proving that the $\times$ operator is well-defined for a quotient ring. I looked up a proof and it omits, without any mention, the part I'm ...
1
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1
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31
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Description of the elements of (p(x))\(q(x))
I am studying Rings and i saw a description of the elements of $R[x]/I$ where $I=Rf=(f(x))=\{p(x)\in R[x]: p(x)=f(x)q(x) \text{ for some $q(x)$ } \}$ (suppose that $deg(f)=n$) then
$$R(x)/I=\{c_0+c_1x+...