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Math Help - Direct Sum

  1. #1
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    Direct Sum

    Let R and S be commutative rings. Prove that the set of all ordered pairs (r,s) such that r is in R and s is in R can be given a ring structure by defining (r1,s1)+(r2,s2)=(r1+r2,s1+s2) and (r1,s1)*(r2,s2)=(r1*r2,s1*s2).


    Let (r1,s1)+(r2,s2)=(r1+r2,s1+s2) and (r1,s1)*(r2,s2)=(r1*r2,s1*s2).

    Would I just show we have an abelian group under addition, multiplication is associative and commutative, there is a multiplicative identity elementm and distributive laws hold?
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  2. #2
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    Yes. In other words, that the object given satisfies the definition of a ring.
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  3. #3
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    I'm a little confused on finding the multiplicative identity element.
    (r1,s1)*(r2,s2)=(r1*r2,s1*s2).
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  4. #4
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    Never mind figured it out.
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