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(Solved): all parts with detail explanation Show if \( Z x Z \) is a. A ring? A commutative ring? b. An integr ...
all parts with detail explanation
Show if \( Z x Z \) is a. A ring? A commutative ring? b. An integral domain? c. 'A field? \[ \begin{array}{l} \forall(n, m),\left(n^{\prime}, m^{\prime}\right) \in Z^{2} \\ +:(n, m)+\left(n^{\prime}, m^{\prime}\right)=\left(n m^{\prime}+n^{\prime} m, m m^{\prime}\right) \in Z^{2} \\ \times:(n, m) \times\left(n^{\prime}, m^{\prime}\right)=\left(n n^{\prime}, m m^{\prime}\right) \in Z^{2} \end{array} \]
Expert Answer
Consider a set Z×Z with given operation of +and x a) To show Z×Z is a ring Let x=(n1,m1),y=(n2,m2),z=(n3,m3) 1) x+(y+z)=(n1,m1)+(n2+n3,m2+m3) =(n1+(n2