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Math Help - Abstract Algebra: Rings with zero divisors involving Cartesian Product

  1. #1
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    Abstract Algebra: Rings with zero divisors involving Cartesian Product

    The problem states:
    Let R and S be nonzero rings. Show that R x S contains zero divisors.

    I had to look up what a nonzero ring was. This means the ring contains at least one nonzero element.

    R x S is the Cartesian Product so if we have two rings R and S
    If r1 r2 belong to R and s1 s1 belong to S

    (r1, s1) + (r2,s2) = (r1+r2, s1 + s2)
    I am using * to denote multiplication here.
    (r1, s1)*(r2,s2) = (r1r2,s1s2)

    Since we are talking about zero divisors I am going to need the definition of multiplication in the Cartesian Product.

    I mean i going to need that (r1r2, s1s2 ) = (0,0)

    so r1r2 = 0 and s1s2 = 0 . But neither r1 = 0 = r2 or s1 = 0 = s2

    Can anyone help?
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  2. #2
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    Quote Originally Posted by jamestrodden View Post
    The problem states:
    Let R and S be nonzero rings. Show that R x S contains zero divisors.

    I had to look up what a nonzero ring was. This means the ring contains at least one nonzero element.

    R x S is the Cartesian Product so if we have two rings R and S
    If r1 r2 belong to R and s1 s1 belong to S

    (r1, s1) + (r2,s2) = (r1+r2, s1 + s2)
    I am using * to denote multiplication here.
    (r1, s1)*(r2,s2) = (r1r2,s1s2)

    Since we are talking about zero divisors I am going to need the definition of multiplication in the Cartesian Product.

    I mean i going to need that (r1r2, s1s2 ) = (0,0)

    so r1r2 = 0 and s1s2 = 0 . But neither r1 = 0 = r2 or s1 = 0 = s2

    Can anyone help?
    (r,0)\cdot(0,s)=(0,0)\,,\,\,0\neq r\in R\,,\,0\neq s\in S

    Tonio
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