1. ## Rings and Ideal

Help me in

Q1 : give an example of not commutative ring and did not have identity.

Q2: Suppose $\displaystyle R$ is ring and $\displaystyle I_{1},I_{2}$are Ideals in $\displaystyle R$ such that :$\displaystyle I_{1}\bigcap I_{2}=\{0\}$
Prove is:$\displaystyle R=I_{1}\bigoplus I_{2}$

2. Originally Posted by nice rose
Help me in

Q1 : give an example of not commutative ring and did not have identity.

The set of all polynomials over the integer numbers with even free coefficient, wrt the usual operations of sum and multiplication of polynomials.

Q2: Suppose $\displaystyle R$ is ring and $\displaystyle I_{1},I_{2}$are Ideals in $\displaystyle R$ such that :$\displaystyle I_{1}\bigcap I_{2}=\{0\}$
Prove is:$\displaystyle R=I_{1}\bigoplus I_{2}$

As stated the problem is false big time: just take $\displaystyle I_1:=\{0\}=$ the zero ideal to get a contradiction, with $\displaystyle R$ any nonzero ring and $\displaystyle I_2$ any non-trivial proper ideal.

Check carefully the conditions...

Tonio

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