sppose that (R,+,.)be aring and we have the ring (R×Z,+^(′ ) ,.^′ ) prove that (R×0,+^′ ,.′) it was ideal in (R×Z,+^′ ,.^′ ) and prove (0×Z,+,.)be isomorphic in (Z,+,.) and if a∈R identity element (a^2 =a)prove that (−a,1)be identity element in the ring (R×Z,+^′ ,.^′ ) pleas sir help me am neding this pleas?


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Question Number 78342 by mhmd last updated on 16/Jan/20

$s p p o s e t h a t (R, +, .) b e a r i n g a n d w e h a v e t h e r i n g (R \times Z, +^{^{'}}, .^{^{'}}) p r o v e t h a t (R \times 0, +^{^{'}}, .^{'}) i t w a s i d e a l i n (R \times Z, +^{^{'}}, .^{^{'}})$ $a n d p r o v e (0 \times Z, +, .) b e i s o m o r p h i c i n (Z, +, .)$ $a n d i f a \in R i d e n t i t y e l e m e n t (a^{2} = a) p r o v e t h a t (- a, 1) b e i d e n t i t y e l e m e n t i n t h e r i n g (R \times Z, +^{^{'}}, .^{^{'}})$ $p l e a s s i r h e l p m e a m n e d i n g t h i s p l e a s ?$
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