The Image of an Ideal Under a Surjective Ring Homomorphism is an Ideal

Problem 532

Let $R$ and $S$ be rings. Suppose that $f: R \to S$ is a surjective ring homomorphism.

Prove that every image of an ideal of $R$ under $f$ is an ideal of $S$.
Namely, prove that if $I$ is an ideal of $R$, then $J=f(I)$ is an ideal of $S$.

Ring Homomorphisms and Radical Ideals
Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism.
Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively.
(a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$.
(b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$
(c) Suppose that $f$ is […]

The Inverse Image of an Ideal by a Ring Homomorphism is an Ideal
Let $f:R\to R'$ be a ring homomorphism. Let $I'$ be an ideal of $R'$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.
Proof.
To prove $I=f^{-1}(I')$ is an ideal of $R$, we need to check the following two […]

The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.
Let
\[R=\left\{\, \begin{bmatrix}
a & b\\
0& a
\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\]
Then the usual matrix addition and multiplication make $R$ an ring.
Let
\[J=\left\{\, \begin{bmatrix}
0 & b\\
0& 0
\end{bmatrix} […]

The Preimage of Prime ideals are Prime Ideals
Let $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$.
Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.
Proof.
The preimage of an ideal by a ring homomorphism is an ideal.
(See the post "The inverse image of an ideal by […]

A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring
Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.
Let $I$ be the subset of $R$ defined by
\[I:=\{ f(x) \in R \mid f(1)=0\}.\]
Then prove that $I$ is an ideal of the ring $R$.
Moreover, show that $I$ is maximal and determine […]

Three Equivalent Conditions for a Ring to be a Field
Let $R$ be a ring with $1$. Prove that the following three statements are equivalent.
The ring $R$ is a field.
The only ideals of $R$ are $(0)$ and $R$.
Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.
Proof. […]

Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$
Let
\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\]
be an ideal of the ring
\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\]
Then determine the quotient ring $\Z[\sqrt{10}]/P$.
Is $P$ a prime ideal? Is $P$ a maximal ideal?
Solution.
We […]

[…] Since the natural projections are surjective ring homomorphisms, the images $I$ and $J$ are ideals in $R$ and $S$, respectively. (see the post The Image of an Ideal Under a Surjective Ring Homomorphism is an Ideal.) […]

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[…] Since the natural projections are surjective ring homomorphisms, the images $I$ and $J$ are ideals in $R$ and $S$, respectively. (see the post The Image of an Ideal Under a Surjective Ring Homomorphism is an Ideal.) […]