Thread: If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai....

1. If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai....

Is true this statements ??
a) If A is a ring, prove that A-module, A[x] is isomorphic with directs sum of Ai. were Ai=A , for i=1,2,3...

, $\displaystyle$A[x] \simeq \Sigma_{i \epsilon N}A_i how A-module.

2. Re: If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai..

Originally Posted by john27
Is true this statements ??
a) If A is a ring, prove that A-module, A[x] is isomorphic with directs sum of Ai. were Ai=A , for i=1,2,3...

, $\displaystyle$A[x] \simeq \Sigma_{i \epsilon N}A_i how A-module.
A module $\displaystyle M$ over an underlying ring $\displaystyle A$ is said to be free if it has a A-basis. Can you write down a basis for $\displaystyle A[x]$? Can you see why a free A-module is necessarily a direct sum of copies of A?

Hint: Look at the definition of direct sums and see how it corresponds to the definition of a basis.

As an aside, we usually write $\displaystyle \oplus_{i\in \mathbb{N}} A_i$ instead of $\displaystyle \sum_{i\in \mathbb{N}}A_i$

3. Re: If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai..

yes or no ?

Are this A-modules isomorphic :$\displaystyle$A[x] \simeq \oplus _{i \epsilon N} A ??