# 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.

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### Is the homomorphic image of a ring an ideal of the co domain? [closed]

Q. If f be a homomorphism from a ring R into a ring R'. Then show that f(R) is an ideal of R'. As per my knowledge, it is not possible. I want a very clear idea about this question and the solution. ...
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### How to break symmetry of a polynomial ideal to simplify Groebner basis?

I have an ideal $I$ generated by a set of polynomials $\{ p_i \}$. There are some variable permutations to which the ideal is symmetric. By this I mean (apologies if there is a standard term for this) ...
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### Ideal $\langle x_1^2 - x_1 , \ldots , x_n^2 - x_n \rangle$ radical?

Consider the ideal generated by the Boolean constraints $$P = \langle x_1^2 - x_1 , \ldots , x_n^2 - x_n \rangle.$$ Is $P$ a radical ideal? A few attempts. The above statement is supposed to be true ...
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### Nilpotent Lie-Algebra $g$: $g^{i+1} ⊆ g^i$ ideal in $g$?

Assume $g$ to be a nilpotent Lie-Algebra. Nilpotency means that we can find an index $n$ such that: $g^n = \{0\}$ for the series defined as: $g^0 = g$ $g^{i+1} = \operatorname{span}\{[g,g^i]\}$ Why is ...
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### Nonunital non commutative ring with 3 ideals...

It is well known that if a (unital commutative) ring A has only three ideals ({0}, J, A), then the quotient A/J is a field. But, what can we conclude about A/J if A is not commutative nor unital but ...
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### Equivalent definition for minimal ideals for commutative rings

Background The following post on minimal ideals is a continuation and a counterpart to the following post on maxmial ideals. The quoted materials are taken from the following sources; Fundamentals of ...
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### How Should I show that these $k$-algebras are not Isomorphic?

Question Show that the $k$-algebras $k[x,y]/\langle xy \rangle$ and $k[x,y]/\langle xy-1 \rangle$ are not isomorphic. Attempt At first, I thought $xy=0$. This would mean both $x$ and $y$ are zero ...
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Question Let $f_1,…,f_s$ be homogeneous polynomials of total degrees $d_1<d_2\leq …\leq d_s$ and let $I=\langle f_1,\ldots,f_s\rangle\subseteq k$. Show that if $g$ is another homogeneous polynomial ...
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### Universal property definition of an ideal generated by a subset?

I'm puzzled by the definition of ideals generated by a subset of a ring in Aluffi, Algebra: Ch 0. The previous chapter on groups is (for an algebra book of this level) quite categorical in spirit. In ...
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### Question about showing $(x,y)$ is a maximal ideal of $\Bbb{Q}[x,y]/F[x,y]$ [duplicate]

Background Theorem 1: Let $M$ be an ideal in a commutative ring $R$ with identity $1_R$. Then $M$ is a maximal ideal if and only if the quotient ring $R/M$ is a field. Exercsie 1: Prove that $(x)$ ...
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### Binomial theorem for ideals

I was proving the statement that if $I$ and $J$ solvable ideals of Lie algebra $L$, then $I + J$ is a solvable ideal of $L$. The proof is we know $$(I+J)/J\cong I/I\cap J.$$ Since $I,J$ are solvable ...
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### Primitive idempotent and bilateral ideals

I'm trying to show for my algebra class that in a semisimple ring with unity $R$ (not necessarily commutative), every primitive idempotent element must belong to a minimal two-sided ideal. Here, by ...
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### Being explicit about the kernel of the map $R[x,y]\to \frac{R}{I}[x,y]$ and coefficients of $\frac{R}{I}[x,y]$

The following is taken from the text University Algebra by: N.S Gopalkrishnan Background Exercise 17: Let $I$ be an ideal of a commutative ring $R$ and let $I[x,y]$ consist of those polynomials with ...
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### If $R$ is an integral domain, $I$ is an ideal of $R$, and $0\neq f: I \to R$ is an $R$-module homomorphism, can we conclude that $f$ is injective?

If $R = \mathbb{Z}$, $0 \neq I \unlhd \mathbb{Z}$, and $0 \neq f: I \to \mathbb{Z}$ is an arbitrary $\mathbb{Z}$-module homomorphism, then $f$ must be injective. This leads to the question: If $R$ is ...
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### $R = \mathbb R[X,Y]/(XY - 1)$ and $I$ be the ideal of $R$ generated by the image of the element $X - Y$ in $R$. Describe $R/I$

Let $R = \mathbb R[X,Y]/(XY - 1)$ ($\mathbb R$ is the set of real numbers) and I be the ideal of R generated by the image of the element X - Y in R. I want to find a way to describe R/I, i.e. find a ...
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### Mistake in Proof "Every unique factorization domain is a principal ideal domain"

While doing my Algebra HW, I "proved" that every unique factorization domain (UFD) is a principal ideal domain (PID). I know that this is not true, however I fail to see where exactly is ...
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### If $J$ is a two-sided ideal of $k$-algebra $A\otimes_k B$, then $I=J\cap B$ is a two-sided ideal of $B$.

Let $A$ and $B$ be finite dimensional $k$-algebra, where $k$ is a field. If $J$ is an two sided ideal of $k$-algebra $A\otimes_k B$, consider $I=J\cap B$, I stuck with proving that I is an two sided ...
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### Show that $M_n(P)$ is a prime ideal of $M_n(R)$.

Let $R$ be a ring and $P$ be a prime ideals of $R$. Then $M_n(P)$ is a prime ideal of $M_n(R)$. One proof of this I know is by using the fact that any ideal of $M_n(R)$ is of the form $M_n(I)$, ...
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