List questions from ideals

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Prove that factor modules are isomorphic.

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Ideals of the quotient ring of A

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List Question

Prime ideals of infinite depth in Noetherian rings

Prove that factor modules are isomorphic.

$I+J=R$ and $r+s=1, r\in I,s\in J$ then $sx+ry\in IJ\Rightarrow x\in I$ and $y\in J$

fraction field of polynomial ring that is a finite extension of the base field

Is the mentioned basis a Gröbner basis?

Proving that S/I is a vector space

The monomials not inside $\operatorname{in}_<(I)$ form a K-basis inside the quotient ring

Polynomial ring, prime ideal, factor ring

Maximal ideals of $R[x_1,\ldots,x_n]$ that is $R$ is a commutative rings with identity

Showing that for every monomial $x^u\in\operatorname{in}_{<}(I)$, there exists $f\in I$ s.t. $\operatorname{in}_<(f)=x^u$

$M$ is maximal, $P$ is prime but not maximal

Bijection between sets of ideals

How to check if a polynomial is inside an ideal using a Groebner basis

If $I=\langle 12 \rangle$, then $\text{Rad}(I)=\langle 6\rangle$

Ideals of the quotient ring of A

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