Show that a ring is commutative if it has the property that ab = ca implies b = c when a != 0.

foass77W 2020-10-19 Answered
Show that a ring is commutative if it has the property that ab = ca implies b = c when a0.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

AGRFTr
Answered 2020-10-20 Author has 95 answers
Since R is ting, therefore by propery of ring we have,
a(ba)=(ab)a for all a,bR
Now, applying given condition on the above property, we get
ba=ab for all a,bR
Therefore, R is cpmmuttive ring
Hene, proved
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-01-19
Let R be a commutative ring. Prove that HomR(R,M) and M are isomorphic R-modules
asked 2022-01-07
Let * be a binary operation on the set of real numbers R defined as follows:
ab=a+b3(ab)2, where a,bR
- Prove that * is commutative but not associative algebraic operation on R.
- Find the identity element for * .
- Show that 1 has two inverses with respect to *.
asked 2020-11-02
Let R be a commutative ring with identity and let I be a proper ideal of R. proe that RI is a commutative ring with identity.
asked 2022-05-11
Let x not be quasinilpotent, so l i m n | | x n | | 1 / n = λ 0. Let π ( x ) = x ¯ , where π ( x ) : A A / r a d ( A ) is the canonical quotient map. Suppose that | | x ¯ n | | 1 / n = 0. Then it's spectrum σ ( x ¯ ) = 0, so x ¯ is not invertible in A / r a d ( A ), and thus generates a proper ideal in A / r a d ( A ). So then π 1 ( x ¯ ) generates a proper ideal in A containing r a d ( A ).
From here, if r a d ( A ) is maximal I think I'd have a contradiction, but I don't know if that's true. If not, does anyone have another strategy I could try?
asked 2020-11-29
Give an example of a commutative ring without zero-divisors that is not an integral domain.
asked 2022-01-04
Prove, that the vector Space Hat (n; F) with the multipliсation
AB=ABBA is a F-algebra (algebra over a field F) is such an algbera associative, commutative, untiary?
asked 2022-06-14
Form the free algebra on a set which is in bijection with S
A i s : s S
and impose the relation that i s is a two-sided inverse of s S for each s S:
S 1 A := A i s : s S s i s 1 , i s s 1 : s S .
Then define ϕ A S 1 A by letting ϕ ( a ) be the image of a in S 1 A. By definition ϕ ( S ) consists of the units of S 1 A.
What I am confused about is what is precisely meant here by
Form the free algebra on a set which is in bijection with A i s : s S .
What is the definition of free algebra over a non-commutative ring A?

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question