# Let R be a commutative ring. If I and P are idelas of R with P prime such that I !sube P, prove that the ideal P:I=P Question
Commutative Algebra Let R be a commutative ring. If I and P are idelas of R with P prime such that $$\displaystyle{I}!\subseteq{P}$$, prove that the ideal $$\displaystyle{P}:{I}={P}$$ 2020-11-28
The ideal quotient of $$\displaystyle{P}{\quad\text{and}\quad}{I}{i}{s}{P}:{I}={\left\lbrace{x}\in{R}:{x}{I}\subset{P}\right\rbrace}$$ which is again an ideal of R.
Given that P is a prime ideal.
To prove that $$\displaystyle{P}:{I}={P}$$
We prove this set equivalence by proving that one set is the subset of other.
For, let $$\displaystyle{x}\in{P}$$
Then $$\displaystyle{x}{I}\subset{P}$$ since P is an ideal.
Therefore, $$\displaystyle{P}\subset{P}:{I}$$
Conversely, let $$\displaystyle{x}\in{P}:{I}$$
Then $$\displaystyle{x}{I}\supset{P}$$. That is $$\displaystyle{x}{y}\in{P}$$ , for every $$\displaystyle{y}\in{I}.$$
Since $$\displaystyle{I}!\supset{P}$$, there exists y in IZSK such that $$\displaystyle{y}\notin{I}.$$ But $$\displaystyle{x}{y}\in{P}.$$
Also since P is a prime ideal, x must be $$\displaystyle\in{P}.$$
Hence $$\displaystyle{x}\in{P}$$ which implies that $$\displaystyle{P}\supset{P}:{I}$$
Hence proved.

### Relevant Questions Suppose that R and S are commutative rings with unites, Let PSJphiZSK be a ring homomorphism from R onto S and let A be an ideal of S.
a. If A is prime in S, show that $$\displaystyle\phi^{{-{{1}}}}{\left({A}\right)}={\left\lbrace{x}\in{R}{\mid}\phi{\left({x}\right)}\in{A}\right\rbrace}$$ is prime $$\displaystyle\in{R}$$.
b. If A is maximal in S, show that $$\displaystyle\phi^{{-{{1}}}}{\left({A}\right)}$$ is maximal $$\displaystyle\in{R}$$. Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is an integral domain. Let R be a commutative ring with identity and let I be a proper ideal of R. proe that $$\displaystyle\frac{{R}}{{I}}$$ is a commutative ring with identity. Suppose that R is a commutative ring and |R| = 30. If I is an ideal of R and |I| = 10, prove that I is a maximal ideal. Let R be a commutative ring. Prove that $$\displaystyle{H}{o}{m}_{{R}}{\left({R},{M}\right)}$$ and M are isomorphic R-modules If R is a commutative ring with unity and A is a proper ideal of R, show that $$\displaystyle\frac{{R}}{{A}}$$ is a commutative ring with unity. Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set:
$$\displaystyle{P}{\left({A}\right)}={\left\lbrace{x}{\mid}{x}\subseteq{A}\right\rbrace}$$
Show that symmetric deference operation on P(A) define by the formula
$$\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)},{x}\in{P}{\left({A}\right)},{y}\in{p}{\left({A}\right)}$$
(where $$\displaystyle{y}^{{c}}$$ is the complement of y) the following statement istrue:
The algebraic operation o+ is commutative and associative on P(A). If A and B are ideals of a commutative ring R with unity and A+B=R show that $$\displaystyle{A}\cap{B}={A}{B}$$ Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in $$\displaystyle{R}\oplus{S}$$ if and only if a or b is a zero-divisor or exactly one of a or b is 0. Let R be a commutative ring with unity and a in R. Then $$\displaystyle{\left\langle{a}\right\rangle}={\left\lbrace{r}{a}:{r}\in{R}\right\rbrace}={R}{a}={a}{R}$$