Question

Can ring homomorphisms be characterized as ring maps such that preimage of any ideal is an ideal?

score 705 · Answer 1 · 2018-02-26 14:31:55

705

Views

Can ring homomorphisms be characterized as ring maps such that preimage of any ideal is an ideal?

Published on 26 Feb 2018 - 14:31

score 767 · Answer 2 · 2018-03-01 19:33:07

767

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Definition of Module Structure Induced by Ring Homomorphism

Published on 01 Mar 2018 - 19:33

score 82 · Answer 3 · 2018-03-18 20:51:43

82

Views

How do I verify if $\phi : \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{36}$ is well-defined?

Published on 18 Mar 2018 - 20:51

score 84 · Answer 4 · 2026-03-22 20:50:26

84

Views

Substitution on $\mathbb{R}$ as isomorphisms

Published on 22 Mar 2026 - 20:50

score 1.9k · Answer 5 · 2018-03-27 21:25:22

1.9k

Views

For ring homomorphism $\phi:R\to S$, prove that $\phi(R)$ is a subring of $S$.

Published on 27 Mar 2018 - 21:25

score 2.8k · Answer 6 · 2018-03-29 20:06:34

2.8k

Views

For ring homomorphism $\phi:R\to S$, prove that if $R$ is a field and $\phi(R)\neq \left \{0_{S} \right \}$, then $\phi(R)$ is a field.

Published on 29 Mar 2018 - 20:06

score 2.5k · Answer 7 · 2026-03-22 15:01:12

2.5k

Views

For ring homomorphism $\phi:R\to S$, prove that $\phi$ is an isomorphism iff $\phi$ is onto and $ker(\phi)=\left \{0_{R} \right \}$.

Published on 22 Mar 2026 - 15:01

score 33 · Answer 8 · 2018-03-30 14:48:47

33

Views

$k$ be an algebraically closed field , $R=M(n,k)$ , then $k \cong End_R (k^n)$

Published on 30 Mar 2018 - 14:48

score 3.3k · Answer 9 · 2018-03-30 22:17:13

3.3k

Views

Find all homomorphisms $\phi: \Bbb{Z}/6\Bbb{Z} \to \Bbb{Z}/15\Bbb{Z}$.

Published on 30 Mar 2018 - 22:17

score 126 · Answer 10 · 2018-04-02 16:17:51

126

Views

On the kernel of the surjective ring homomorphism $R[X] \to R[ab^{-1}]$ taking $X$ to $ab^{-1}$

Published on 02 Apr 2018 - 16:17

score 141 · Answer 11 · 2018-04-03 04:34:37

141

Views

Proving that $D^{-1}\mathbb{Z}\cong \mathbb{Z}[x]/(px-1)$

Published on 03 Apr 2018 - 4:34

score 1.1k · Answer 12 · 2018-04-03 11:33:57

1.1k

Views

Example of non surjective ring homomorphisim where im$(\phi)$ is not an ideal

Published on 03 Apr 2018 - 11:33

score 59 · Answer 13 · 2026-03-22 08:12:03

59

Views

Proof this factor ring is field.

Published on 22 Mar 2026 - 8:12

score 374 · Answer 14 · 2026-03-22 03:28:45

374

Views

Basis for $R[x]/(f)$ as an $R$-module

Published on 22 Mar 2026 - 3:28

score 79 · Answer 15 · 2018-04-07 22:31:01

79

Views

Show that the ring homomorphism $\phi:R\to R/I$ defined by $\phi(r)=r+I$ is well-defined.

Published on 07 Apr 2018 - 22:31

List Question

Can ring homomorphisms be characterized as ring maps such that preimage of any ideal is an ideal?

Definition of Module Structure Induced by Ring Homomorphism

How do I verify if $\phi : \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{36}$ is well-defined?

Substitution on $\mathbb{R}$ as isomorphisms

For ring homomorphism $\phi:R\to S$, prove that $\phi(R)$ is a subring of $S$.

For ring homomorphism $\phi:R\to S$, prove that if $R$ is a field and $\phi(R)\neq \left \{0_{S} \right \}$, then $\phi(R)$ is a field.

For ring homomorphism $\phi:R\to S$, prove that $\phi$ is an isomorphism iff $\phi$ is onto and $ker(\phi)=\left \{0_{R} \right \}$.

$k$ be an algebraically closed field , $R=M(n,k)$ , then $k \cong End_R (k^n)$

Find all homomorphisms $\phi: \Bbb{Z}/6\Bbb{Z} \to \Bbb{Z}/15\Bbb{Z}$.

On the kernel of the surjective ring homomorphism $R[X] \to R[ab^{-1}]$ taking $X$ to $ab^{-1}$

Proving that $D^{-1}\mathbb{Z}\cong \mathbb{Z}[x]/(px-1)$

Example of non surjective ring homomorphisim where im$(\phi)$ is not an ideal

Proof this factor ring is field.

Basis for $R[x]/(f)$ as an $R$-module

Show that the ring homomorphism $\phi:R\to R/I$ defined by $\phi(r)=r+I$ is well-defined.

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