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

• Jun 6th 2013, 06:21 AM
john27
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...

, $A[x] \simeq \Sigma_{i \epsilon N}A_i$ how A-module.
• Jun 6th 2013, 10:35 PM
Gusbob
Re: If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai..
Quote:

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...

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

A module $M$ over an underlying ring $A$ is said to be free if it has a A-basis. Can you write down a basis for $A[x]$? 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 $\oplus_{i\in \mathbb{N}} A_i$ instead of $\sum_{i\in \mathbb{N}}A_i$
• Jun 15th 2013, 03:13 PM
john27
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 : $A[x] \simeq \oplus _{i \epsilon N} A$ ??