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Math Help - Integral domain and unit of a ring

  1. #1
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    Integral domain and unit of a ring

    Lt R be a commutative ring not neccesarily an integral domain. Determine the truth or falsity of the following statements. In each case, give a proof or find a counter-example

    Let x, y belongs to R
    (i) x.y in R* ---> x in R* , y in R* (R* = R\{0})
    (ii) x.y in U(R) ---> x in U(R), y in U(R)
    (iii) x.y in R*\ U(R) ---> x in R*\ U(R), y in R*\U(R)

    This is my attempt

    (i)If x.y in R* then we have x.y not equal to 0 implies that x is nonzero and y is nonzero. So, the statement is true

    (ii)I think this statement is false but I dont know how to prove it

    (iii)This part I dont know

    thank you
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  2. #2
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    (i)If x.y in R* then we have x.y not equal to 0 implies that x is nonzero and y is nonzero. So, the statement is true
    Yes

    (ii)I think this statement is false but I dont know how to prove it
    Well x.y\in U(R) means there is a\in R such that a.(x.y)=1=(x.y).a\ . The multiplication in a ring is associative; so what can you deduce from these equalities?

    (iii)This part I dont know
    Try to see what happens if you take y=1.
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  3. #3
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    (ii)I think this statement is false but I dont know how to prove it

    what can you deduce from these equalities?

    It implies that xy is the inverse of a, and since xy in U(R) then x in U(R) and y in U(R). Hence the statement is true
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  4. #4
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    Quote Originally Posted by knguyen2005 View Post
    (ii)I think this statement is false but I dont know how to prove it

    what can you deduce from these equalities?

    It implies that xy is the inverse of a, and since xy in U(R) then x in U(R) and y in U(R). Hence the statement is true
    I don't think your argument is valid here. I think you're supposed to deduce that x(ya)=1=(ya)x and y(xa)=1=(xa)y. You can do this since R is a commutative ring. From those equalities, you get x,y \in U(R)
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by clic-clac View Post
    Try to see what happens if you take y=1.
    You can't take y=1, as this is a unit. We are looking at the non-zero ring element which are not units.
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  6. #6
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    What we want to prove is the assertion is wrong, i.e. x.y can be non invertible with x or y invertible.

    With y=1, it becomes clear that whenever there are non zero non invertible elements in R, then the implication is false.


    Edit: Quite close indeed
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  7. #7
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by clic-clac View Post
    What we want to prove is the assertion is wrong, i.e. x.y can be non invertible with x or y invertible.

    With y=1, it becomes clear that whenever there are non zero non invertible elements in R, then the implication is false.
    I know - I'm having a slow evening, and I couldn't get to my laptop in time to change it!

    EDIT: Although, by the timing of this post it looks like I was pretty close!
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