All Questions
Tagged with ideals group-theory
64
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Are there any subsets $I$ of $\mathbb{Z}$ containing $0$ & closed under multiplication by elements of $\mathbb{Z}$, but not an ideal of $\mathbb{Z}$?
Are there any subsets $I$ of $\mathbb{Z}$ containing $0$ & closed under multiplication by elements of $\mathbb{Z}$, but not an ideal of $\mathbb{Z}$?
This was an exam question at my university ...
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I'm confused by the last step when showing that $\Phi$ is well defined in the proof of 2nd isomorphism theorem for rings? [closed]
$S$ is a subring of $R$, $J$ is an ideal of $R$, and $\Phi: (S + J) \to S/(S \cap J)$ where $\Phi(a + b) = a + (S \cap J)$. We need to show that $\Phi$ is well defined.
Suppose that $a_1 + b_1 = a_2 + ...
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Does there exist an "arithmetically-defined" group embedded as some subset of all cosets $a\Bbb{Z} + b$? Because we easily have monoids...
Let $A \subset \Bbb{Z}$ be any set closed under taking $\gcd$, then $\{a\Bbb{Z} + u : a \in A, u \in U\}$, where $U$ is any multiplicative subgroup of $\{-1, 1\}$, forms an elementwise monoid under ...
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Find the generator of a cyclic group $(\ ^{\mathbb{Z}_{5}[x]}\!/\!_{(g(x))})^{*}$
Consider the field $\mathbb{Z}_{5}$ and the polynomial ring $\mathbb{Z}_{5}[x]$ . $g(x) =x^{3}+x+1$ is an irreducible polynomial in $\mathbb{Z}_{5}[x]$ and hence $(g(x))$ is a prime and maximal idea. ...
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Similarities Between Rings and Groups: Ideals and Normal Subgroups
I've been pondering the parallels between rings and groups and the commonalities shared by ideals and normal subgroups. There's a theorem stating that if $R$ is a ring and it happens to be a field, ...
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Characterizations of divisors of the order of an element.
For prime numbers $p$ consider the two conditions
$({\rm A})\ \ \forall\:\! k\geq0\!:\ (-2)^k+1\not\equiv0\,\pmod{\!p}$
$({\rm B})\ \ \exists\:\! k\geq0\!:\ \ 2^{2k+1}+1\equiv0\,\pmod{\!p}$
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- 253
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Prove that $R/U$ forms a group under addition defined in $R.$ ($R$ and $U$ is a ring and ideal respectively)
If $R$ is a ring and $U$ be the (two-way) ideal of a ring. Then we define, $R/U$ to be the set of all cosets of $R$ under addition. Prove that $R/U$ forms a group under addition defined in $R.$
I ...
- 3,073
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Maximal ideal of $F^E$
Let $F$ be a field and $E$ a set, we consider the ring $F^E$ (ring of maps from $E$ to $F$)
I was able to prove that if, $ E $ is finite then all the maximal ideals of $F^E$ are exactly:
$$M_a = \{...
- 339
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Can we come up with a disjoint union of a subsets of the group $\Bbb{Z}$ such that they do not equal the cosets of a subgroup, yet they form a group?
If this applies to $\Bbb{Z}$ it probably will work for other groups $G$, however, for simplicity and because I'm interested in integers & their primes, let's work with $G = \Bbb{Z}$.
Anyway, we ...
2
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1
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What kind of structure is $\mathbb{Z}[i]/\langle 2+2i\rangle$?
Consider the factor ring $R=\mathbb{Z}[i]/\langle 2+2i\rangle$, where $\langle 2+2i\rangle$ is the ideal of the Gaussian integers such that for all $z\in\mathbb{Z}[i]$, $z(2+2i)\in\langle 2+2i\rangle$....
- 1,337
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Let $I$ be a two-sided ideal of a ring $R$. Show: An abelian group $M$ is a left $(R/I)$-module iff $M$ is a left $R$-module annihilated by $I$. [duplicate]
I have so far figured out neither of the directions of the proof. I have a very specific point I am stuck at right now. I am trying to prove the "$\Leftarrow$"-direction at the moment, so I ...
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How to compute the elements of an Ideal of a polynomial ring?
I am reading about polynomial quotient rings and the text starts with
Let $\Bbb R[x]$ denote the ring of polynomials with real coefficients and let $\langle x^2 + 1\rangle $ denote the principal ...
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Show that $\Bbb Z_5[x]/\langle x^2+1\rangle$ is not an integral domain
Show that $\Bbb Z_5[x]/\langle x^2+1\rangle$ is not an integral domain.
My approach:
Since $x^2+1 = (x-2)(x-3)$ is in $\Bbb Z_5[x]$ it is reducible in $\Bbb Z_5[x]\implies\langle x^2+1\rangle $ ideal ...
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Ideals versus normal subgroups
Let $\phi : R \to R'$ be a ring homomorphism with $S$ a subring of $R$. Let $I, J$ be ideals of $R$ with $I \subseteq J \subseteq R$.
There is a one-to-one correspondence between the ideals of $R$ ...
- 1,751
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Ideal generated by x and y in the ring of polynomials with real coefficients
Consider $\mathbb{R}[x,y]$ and $I,J$ ideals of $\mathbb{R}[x]$ given by $I=J =(x,y)$. In lecture, my professor told us that $x^2+y^2 \neq ij$ for some $i,j \in (x,y)$. Although this is somewhat ...
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