Question

Let $R$ be a commutative ring with identity $1.$ Let $A$ and $B$ be two maximal ideals of $R.$ Prove that $AB=A\cap B.$

score 75 · Answer 1 · 2026-04-01 15:01:15

75

Views

Let $R$ be a commutative ring with identity $1.$ Let $A$ and $B$ be two maximal ideals of $R.$ Prove that $AB=A\cap B.$

Published on 01 Apr 2026 - 15:01

score 64 · Answer 2 · 2023-12-18 11:45:41

64

Views

Theorem 7, Section 3.4 of Hungerford’s Algebra

Published on 18 Dec 2023 - 11:45

score 126 · Answer 3 · 2026-03-26 22:55:34

126

Views

If $R$ is a ring such that $\forall x\in R, x^n=x$ for some $n>1$ then when $P$ is prime, why is $R/P$ finite?

Published on 26 Mar 2026 - 22:55

score 36 · Answer 4 · 2023-12-19 01:41:11

If an ideal $I$ in a unital commutative ring equals the intersection of some prime ideals, then it equals the intersection of all prime ideals.

score 79 · Answer 5 · 2026-03-26 21:26:07

79

Views

Is the span of all nilpotent ideals also a nilpotent ideal?

Published on 26 Mar 2026 - 21:26

score 115 · Answer 6 · 2026-04-01 15:02:54

115

Views

If every prime ideal of $A$ is maximal, then the chain ${}\dots \leq x^2 A \leq x A$ stabilizes, for all $x \in A$.

Published on 01 Apr 2026 - 15:02

score 58 · Answer 7 · 2023-12-27 21:40:16

58

Views

Sum of two ideals, their product, and intersection

Published on 27 Dec 2023 - 21:40

score 83 · Answer 8 · 2023-12-29 00:34:11

83

Views

Do we get a field when we form the quotient ring of polynomials over $\mathbb{Q}$ modulo polynomials that annihilate $\sqrt{2}+\sqrt{3}$?

Published on 29 Dec 2023 - 0:34

score 54 · Answer 9 · 2026-03-28 08:35:33

54

Views

If $I = (xy,(x-y)z)$ is an ideal in $k[x,y,z]$ where $k$ is a field, then $\operatorname{rad} I = (xy,xz,yz)$.

Published on 28 Mar 2026 - 8:35

score 46 · Answer 10 · 2026-03-20 23:43:21

46

Views

The fact that the space of matrix coefficients is a 2-side ideal in $C(G)$ implies Schur orthogonality.

Published on 20 Mar 2026 - 23:43

score 116 · Answer 11 · 2026-03-25 21:48:00

116

Views

A unital commutative ring having a square-zero maximal ideal is local

Published on 25 Mar 2026 - 21:48

score 69 · Answer 12 · 2026-03-28 08:35:37

69

Views

Distributive property of sum and intersection of ideals in a polynomial ring

Published on 28 Mar 2026 - 8:35

score 92 · Answer 13 · 2026-04-01 16:27:08

92

Views

Let $\mathfrak{q}$ be a $\mathfrak{p}$-primary ideal. Does $\mathfrak{p}^{n} \subset \mathfrak{q} \subset \mathfrak{p}^{n-1}$ for some $n$?

Published on 01 Apr 2026 - 16:27

score 109 · Answer 14 · 2024-01-06 14:47:36

109

Views

What is (if any) the generalization of the concept of a basis for ideals of rings?

Published on 06 Jan 2024 - 14:47

score 47 · Answer 15 · 2024-01-08 16:55:23

47

Views

Is there a name to $[R:I]$ in Ring Theory?

Published on 08 Jan 2024 - 16:55

List Question

Let $R$ be a commutative ring with identity $1.$ Let $A$ and $B$ be two maximal ideals of $R.$ Prove that $AB=A\cap B.$

Theorem 7, Section 3.4 of Hungerford’s Algebra

If $R$ is a ring such that $\forall x\in R, x^n=x$ for some $n>1$ then when $P$ is prime, why is $R/P$ finite?

If an ideal $I$ in a unital commutative ring equals the intersection of some prime ideals, then it equals the intersection of all prime ideals.

Is the span of all nilpotent ideals also a nilpotent ideal?

If every prime ideal of $A$ is maximal, then the chain ${}\dots \leq x^2 A \leq x A$ stabilizes, for all $x \in A$.

Sum of two ideals, their product, and intersection

Do we get a field when we form the quotient ring of polynomials over $\mathbb{Q}$ modulo polynomials that annihilate $\sqrt{2}+\sqrt{3}$?

If $I = (xy,(x-y)z)$ is an ideal in $k[x,y,z]$ where $k$ is a field, then $\operatorname{rad} I = (xy,xz,yz)$.

The fact that the space of matrix coefficients is a 2-side ideal in $C(G)$ implies Schur orthogonality.

A unital commutative ring having a square-zero maximal ideal is local

Distributive property of sum and intersection of ideals in a polynomial ring

Let $\mathfrak{q}$ be a $\mathfrak{p}$-primary ideal. Does $\mathfrak{p}^{n} \subset \mathfrak{q} \subset \mathfrak{p}^{n-1}$ for some $n$?

What is (if any) the generalization of the concept of a basis for ideals of rings?

Is there a name to $[R:I]$ in Ring Theory?

Trending Questions

Popular # Hahtags

Popular Questions