# Recent questions in Commutative Algebra

Commutative Algebra

### 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.

Commutative Algebra

### Let R be a commutative ring with unity and a in R. Then $$\displaystyle{\left\langle{a}\right\rangle}={\left\lbrace{r}{a}:{r}\in{R}\right\rbrace}={R}{a}={a}{R}$$

Commutative Algebra

### Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.

Commutative Algebra

### Let a belong to a ring R. Let $$\displaystyle{S}={\left\lbrace{x}\in{R}{\mid}{a}{x}={0}\right\rbrace}$$ . Show that S is a subring of R.

Commutative Algebra

### 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).

Commutative Algebra

### Suppose that R is a ring and that $$\displaystyle{a}^{{2}}={a}$$ for all $$\displaystyle{a}\in{R}{Z}$$. Show that R is commutative.

Commutative Algebra

### Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.

Commutative Algebra

### Let R be a commutative ring. Prove that $$\displaystyle{H}{o}{m}_{{R}}{\left({R},{M}\right)}$$ and M are isomorphic R-modules

Commutative Algebra

### Show that quaternion multiplication is not commutative. That is, give an example to show that sometimes $$\displaystyle\varepsilon\eta$$ does not equal $$\displaystyle\eta\varepsilon$$.

Commutative Algebra

### Let R be a commutative ring. Show that R[x] has a subring isomorphic to R.

Commutative Algebra

### Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.

Commutative Algebra

### 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.

Commutative Algebra

### Let * be a binary operation on set of rational number $$\displaystyle\mathbb{Q}$$ defined as follows: a*b=a+b+2ab, where $$\displaystyle{a},{b}\in\mathbb{Q}$$ a) Prove that * is commutative, associate algebraic operation on $$\displaystyle\mathbb{Q}$$

Commutative Algebra

### 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.

Commutative Algebra

### If A and B are ideals of a commutative ring R with unity and A+B=R show that $$\displaystyle{A}\cap{B}={A}{B}$$

Commutative Algebra

### 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.

Commutative Algebra

### Show that $$\displaystyle{\left(\mathbb{Z}_{{6}}+_{{6}},\times_{{6}}\right)}$$ is a commutative ring. Is $$\displaystyle{\left(\mathbb{Z}_{{6}}+_{{6}},\times_{{6}}\right)}$$ a field?

Commutative Algebra

### List all zero-divisors in $$\displaystyle{Z}_{{20}}$$. Can you see relationship between the zero-divisors of $$\displaystyle{Z}_{{20}}$$ and the units of $$\displaystyle{Z}_{{20}}$$?

Commutative Algebra