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COMMUTATIVE ALGEBRA
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Commutative algebra problems solved
Recent questions in Commutative Algebra
Commutative Algebra
asked 2021-03-07
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
asked 2021-03-02
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
asked 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.
Commutative Algebra
asked 2021-02-11
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
asked 2021-02-05
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
asked 2021-01-31
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
asked 2021-01-27
Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.
Commutative Algebra
asked 2021-01-19
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
asked 2020-12-24
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
asked 2020-12-21
Let R be a commutative ring. Show that R[x] has a subring isomorphic to R.
Commutative Algebra
asked 2020-12-17
Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.
Commutative Algebra
asked 2020-12-17
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
asked 2020-12-16
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
asked 2020-12-14
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
asked 2020-12-09
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
asked 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.
Commutative Algebra
asked 2020-12-03
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
asked 2020-11-30
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
asked 2020-11-29
Give an example of a commutative ring without zero-divisors that is not an integral domain.
Commutative Algebra
asked 2020-11-27
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
\(\displaystyle{P}:{I}={P}\)
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