Question

Question 12: If p is a prime integer, prove that M is a maximal ideal in $\mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}}$, where $\mathbf{M}=\left\{\left.\left(\mathbf{pa}\mathbf{,}\mathbf{b}\right)\right|\mathbf{a}\mathbf{,}\mathbf{b}\in \mathrm{\mathbb{Z}}\right\}$ .

Short answer

Answer:

It is proved that the M is a maximal ideal in$\mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}}$ .

Step-by-step solution

Statement of theorem 6.15 

Theorem 6.15 states that ifis an ideal in a commutative ringR with identity. Then, M would be a maximal idealif and only if the quotient ring R/M would be a field.

Show that M is a maximal ideal in ℤ×ℤ , where  M=pa,ba,b∈ℤ 

First isomorphism theoremstates that if$f:R\to S$is a surjective homomorphism of rings with kernel K . Then, thequotient ring R/K would be isomorphic to S.

Introduce the surjective homomorphism $\phi :\mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}}\to {\mathrm{\mathbb{Z}}}_p;\phi \left(a,b\right)=\left[a\right]$.

Therefore, $\ker \phi =M$ and $\left(\mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}}\right)/M\cong {\mathrm{\mathbb{Z}}}_p$ would be field because of the first isomorphism theorem.

As a result, would be a maximal ideal according to theorem 6.15.

Hence, it is proved that the M is a maximal ideal in$\mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}}$ .