 # Commutative algebra Questions and Answers

Recent questions in Commutative Algebra Khaleesi Herbert 2021-03-02

### Let R be a commutative ring with unity and a in R. Then $⟨a⟩=\left\{ra:r\in R\right\}=Ra=aR$ Maiclubk 2021-02-11

### Let a belong to a ring R. Let $S=\left\{x\in R\mid ax=0\right\}$ . Show that S is a subring of R. Wotzdorfg 2021-02-11

### Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order. Bergen 2021-02-05

### Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set: $P\left(A\right)=\left\{x\mid x\subseteq A\right\}$ Show that symmetric deference operation on P(A) define by the formula $x\oplus y=\left(x\cap {y}^{c}\right)\cup \left(y\cap {x}^{c}\right),x\in P\left(A\right),y\in p\left(A\right)$ (where ${y}^{c}$ is the complement of y) the following statement istrue: The algebraic operation o+ is commutative and associative on P(A). Clifland 2021-01-31

### Suppose that R is a ring and that ${a}^{2}=a$ for all $a\in RZ$. Show that R is commutative. he298c 2021-01-27

### Assume that the ring R is isomorphic to the ring R. alesterp 2021-01-19

### Let R be a commutative ring. Prove that $Ho{m}_{R}\left(R,M\right)$ and M are isomorphic R-modules bobbie71G 2020-12-24

### Show that quaternion multiplication is not commutative. That is, give an example to show that sometimes $\epsilon \eta$ does not equal $\eta \epsilon$. defazajx 2020-12-21

### Let R be a commutative ring. Show that R[x] has a subring isomorphic to R. banganX 2020-12-17

### If R is a commutative ring with unity and A is a proper ideal of R, show that $\frac{R}{A}$ is a commutative ring with unity. CoormaBak9 2020-12-17

### Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors. FizeauV 2020-12-16

### Let $×$ be a binary operation on set of rational number $\mathbb{Q}$ defined as follows: $a\cdot b=a+b+2ab$, where $a,b\in \mathbb{Q}$ a) Prove that $×$ is commutative, associate algebraic operation on $\mathbb{Q}$ Cheyanne Leigh 2020-12-14

### Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in $R\oplus S$ if and only if a or b is a zero-divisor or exactly one of a or b is 0. Tolnaio 2020-12-09

### 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. boitshupoO 2020-12-09

### If A and B are ideals of a commutative ring R with unity and A+B=R show that $A\cap B=AB$ Braxton Pugh 2020-12-03

### Show that $\left({\mathbb{Z}}_{6}{+}_{6},{×}_{6}\right)$ is a commutative ring. Is $\left({\mathbb{Z}}_{6}{+}_{6},{×}_{6}\right)$ a field? snowlovelydayM 2020-11-30

### List all zero-divisors in ${Z}_{20}$. Can you see relationship between the zero-divisors of ${Z}_{20}$ and the units of ${Z}_{20}$? emancipezN 2020-11-29

### Give an example of a commutative ring without zero-divisors that is not an integral domain. Maiclubk 2020-11-27

### Let R be a commutative ring. If I and P are idelas of R with P prime such that $I!\subseteq P$, prove that the ideal $P:I=P$ Trent Carpenter 2020-11-23

### If R is a commutative ring, show that the characteristic of R[x] is the same as the characteristic of R.

In basic terms, commutative algebra represents a complex study of the rings that take place in algebraic number theory. It also relates to solving problems and questions based on algebraic geometry. You will see that there are solutions related to Dedekind rings and various commutative patterns. Commutative algebra solutions will be quite short in most cases, yet one should start with examples that are represented and link them with various questions. Take your time to post your commutative algebra example problem as well and always compare your question to similar questions as it will help you provide the required data. When you are dealing with inequality and graph solution tasks as you are seeking commutative equ