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$\sqrt{I}+\sqrt{J}=R$ implies $I+J=R$

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#abstract-algebra #proof-verification #ideals
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A relation between the Jacobson radicals of a ring and those of a certain quotient ring

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#ring-theory #representation-theory #ideals #maximal-and-prime-ideals
133
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Simplify $(y-x^2)\cap(y^2+2y+x^2)$

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#algebraic-geometry #commutative-algebra #ideals
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$\mathbb{Z}_{(2)}$ has one maximal ideal

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#commutative-algebra #ideals
244
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$I$ is maximal ideal $\implies$ $R/I$ has no proper ideals

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#abstract-algebra #ring-theory #ideals #maximal-and-prime-ideals
51
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No such prime ideal contains $I_1+I_2\implies I_1 $ and $I_2$ are relatively prime

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#commutative-algebra #ideals #maximal-and-prime-ideals
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For a maximal ideal $M$ of $R$ of a commutative ring $R$ ( not necessarily with unity ) , then is $R/M$ a simple ring ?

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#ring-theory #ideals #maximal-and-prime-ideals
445
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Determine all units in $\mathbb{Z}[\omega] := \{a+b\omega\mid a,b\in\mathbf{Z}\}$ where $\omega = \frac{-1 + i \sqrt{3}}{2}$

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#number-theory #ring-theory #algebraic-number-theory #ideals
4.6k
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$Rad(I)$ is an ideal of $R$

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#abstract-algebra #ideals #radicals
831
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Operations with ideals: sum and product

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#abstract-algebra #ideals
64
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Proof of the proposition $V(S)=V(\langle S \rangle )$

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#abstract-algebra #ideals #roots
542
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Algebraic Set-Radical Ideal-Nullstellensatz

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#abstract-algebra #ideals
538
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Algebraic set - Radical Ideal - $Rad(Rad(I))=Rad(I)$

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#abstract-algebra #ideals
196
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$V_1=V(x-y)$ and $V_2=V(x+y)$ are algebraic sets

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#algebraic-geometry #ideals
67
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Why is that the radical ideal?

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#abstract-algebra #ideals

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