Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [ideals]

An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset.

This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ideals in Lie algebras.

An ideal $I$ in a ring $(R,+,\cdot)$ a subset $I\subseteq R$ such that $(I,+)$ is a subgroup of the additive group $(R,+)$ and $r\cdot x,x\cdot r\in I$ whenever $r\in R$ and $x\in I$ (i.e., $I$ is closed under multiplication by arbitrary elements).

This is the most frequent use of the name ideal, but it is used in other areas of mathematics too:

5753 questions
0
votes
2 answers

Why does the ideal generated by x generate polynomials?

This question is at a very low level. Let $R$ be a commutative ring with unity. I am using the standard definition $\langle x \rangle = \{rx: r \in R\}$ for the ideal generated by $x$. It seems like the polynomial $p(x) = x^2 + x$ should not live…
sobrio35
  • 111
0
votes
1 answer

Finding an isomorphism between $\mathbf Z[\sqrt{-5}]\big/(3,1-\sqrt{-5})$ and $\mathbf Z\big/3$

Finding an isomorphism between $\mathbf Z[\sqrt{-5}]\big/(3,1-\sqrt{-5})$ and $\mathbf Z\big/3$ ? In general how does the elements of the ideal $(3,1-\sqrt{-5})$ look like, in the form $(3)\cup(1-\sqrt{-5})$ ?
user257
  • 989
0
votes
1 answer

Is the difference of ideals an ideal?

I am studying ideals and noticed that $I+J$ is an ideal as noted here. However the paper does not discuss $I-J$ so: Is the difference of ideals an ideal?
hhh
  • 5,469
0
votes
2 answers

How to prove that two principal ideals are equal

Background info Provided that the general form of a polynomial is $(a_0X^{0}+...+a_nX^{n})$ where $X$ is an element of the field provided. an ideal generated by an element is the set $(a*r s.t. a\in V, r\in ring)$ Question For $F$ a field, and…
HELP
  • 182
0
votes
1 answer

Why $IJ\subset I\cap J$ (for $I$ and $J$ ideal) whereas if $N$ and $H$ are groups $N,H\leq NH$

Let $N,H$ two subgroup of a group $G$ such that at least one is normal. By Surb answer here, $NH$ is the smallest group that contain $N$ and $H$. But if $I$ and $J$ are ideal, they are also group for $+$, therefore we should have $I\cap J\leq…
Rick
  • 1,707
0
votes
1 answer

Equalities of ideals in Q[x,y]

I'm trying to prove something about polinomyal ideals. So I have to use this proposition: Let $I \subset k[x_{1}, \ldots, x_{n}] $ be an ideal, and let $f_{1}, \ldots, f_{s} \in k[x_{1}, \ldots, x_{n}].$ Then these are equivalent: $(i) f_{1},…
HelaHelheim
  • 169
0
votes
1 answer

Lack of unique factorization of proper ideals in $\mathbb{Z}[\sqrt{-3}]$

I am working on an exercise that asks us to consider the ring $R = \mathbb{Z}[\sqrt{-3}]$ and the ideal $I = (2, 1 + \sqrt{-3})$ in $R$. Part (a) asks to show that $I^2 = (2)I$ but $I \neq (2)$, and part (b) asks to show that $I$ is the unique prime…
Ethan Alwaise
  • 9,286
0
votes
1 answer

Morita equivalence and right and left ideals of a Ring

I have been thinking a bit about Morita equivalence http://en.wikipedia.org/wiki/Morita_equivalence and I would like to know whether it also applies to subrings such as right or left ideals. And, if so, how specifically? Could you illustrate if…
Javier Arias
  • 2,033
0
votes
0 answers

Visual representation of (right or left) ideals of a ring?

I would like to know about the canonical visual representation for ideals in a ring. Particularly, for the two kinds of one-sided ideals,that is, right ideals and left ideals. Is it possible to use Venn diagrams or the like in order to represent…
Javier Arias
  • 2,033
0
votes
2 answers

Show that a collection of functions $I$ is an ideal

Let $R$ be the ring of all continuous function on $[0,1]$, and let $I$ be the collection of functions $f(x)$ in $R$ with $f(1/3)=f(1/2)=0$. Prove that $I$ is an ideal of $R$ but not a prime ideal. I know that when you want to show something is an…
user146269
  • 1,855
1 2 3
4