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Math Help - Subring problem

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    Subring problem

    Let S , T be rings, is S \cap T and  S \cup T also rings?

    I think the second one is yes, but the first one is no, but how do I find an example like that? Thanks.
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    Quote Originally Posted by tttcomrader View Post
    Let S , T be rings, is S \cap T and  S \cup T also rings?

    I think the second one is yes, but the first one is no, but how do I find an example like that?
    Here's an example to think about. Let S denote all integer multiples of 3, and T denote all integer multiples of 5. Then 3∈S and 5∈T. But 3+5=8 which is not a multiple of 3 or 5. So S∪T is not closed under addition.
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    Quote Originally Posted by Opalg View Post
    Here's an example to think about. Let S denote all integer multiples of 3, and T denote all integer multiples of 5. Then 3∈S and 5∈T. But 3+5=8 which is not a multiple of 3 or 5. So S∪T is not closed under addition.
    most ring theorists define a subring S of a unitary ring R to be a unitary ring with 0_S=0_R \neq 1_R=1_S. so, to them, n\mathbb{Z}, \ n \geq 0, is a subring of \mathbb{Z} iff n = 1.


    Edit: ok, for some weird reasons, i thought S and T have to be subrings of a unitary ring! lol the problem is just asking for any rings, which makes things much

    easier and Opalg's example is perfectly fine! my apologies Opalg!
    Last edited by NonCommAlg; October 14th 2008 at 02:49 PM.
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    Quote Originally Posted by NonCommAlg View Post
    most ring theorists define a subring S of a unitary ring R to be a unitary ring with 0_S=0_R \neq 1_R=1_S. so, to them, n\mathbb{Z}, \ n \geq 0, is a subring of \mathbb{Z} iff n = 1.
    Well I'm not a ring theorist and nobody told me they had to be unitary.

    If you want unitary subrings, how about looking at subrings of the 33 matrices over the integers? You could take S to be all those of the form \begin{bmatrix}a&x&0\\0&b&0\\0&0&c\end{bmatrix}, and T to be all those of the form \begin{bmatrix}a&0&0\\0&b&y\\0&0&c\end{bmatrix}. Then S∪T is not closed under addition.

    (I'm sure there are simpler examples that a bona fide ring theorist could provide.)
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    Quote Originally Posted by Opalg View Post
    (I'm sure there are simpler examples that a bona fide ring theorist could provide.)
    S=R[X], \ T=R[Y], as subrings of the polynomial ring R[X,Y]. then S \cup T is not closed under addition (or multiplication).
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