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    • Thread Starter

    Could someone please tell me is these statements are true:

    i) Every Field is a Ring.
    ii) Every Integral Domain is a Ring.
    iii) Every Field is an Integral Domain.
    iv) Every finite Integral Domain is a Field.

    v) Not all Rings are Fields.
    vi) Not all Rings are Integral Domains.
    vii) Not all Integral Domains are Fields.

    Which statements are true and which are false. I am trying to get my head round the connection between them all. Thanks
    • Thread Starter

    Added to this, I have in my notes:

    1)F[x] is an integral domain.
    2)deg(f(x)g(x)) = deg(f(x))+deg(g(x)).
    3) This also holds for R is an ID, but if R is not an ID then it is false.

    Does this mean all ID's have the property of 2) as all ID's are Rings??

    I might be confusing thing here...

    (Original post by adie_raz)
    Basically, all fields are integral domains and all integral domains are rings.

    iv ... err... if a is non-zero then the map f:R->R given by f(x) -> ax is 1-1. As R is finite, f is also onto. So there is some x so that ax=1. So the ring has multiplicative inverses.

    Not all rings are fields. Eg, any ring with zero divisors (the ring of 2x2 real matrices for example).
    Not all rings are IDs. Same counterexample as above.
    Not all IDs are fields. Eg the integers.

    If F is a field then F[x] is an ID. All IDs have property 2 because if f(x) = ax^n + ... + q (a non zero) and g(x) = bx^m + ... + r (b non zero) then f(x)g(x) = abx^(m+n) + ... + qr. In an ID, a and b are non-zero so ab is non-zero. So deg(fg)=m+n=deg(f)+deg(g). If you had polynomials over a non-ID then you could pick a and b to be non-zero but such that ab=0. Then deg(fg) would be less than deg(f)+deg(g).

    All the statements in your first post are true. I'm not certain on (iv) but I'm pretty sure that is true. To help you understand them, try drawing a picture, i.e.

    Since all fields are rings but not all rings are fields, the set of rings strictly contains the set of fields. You can fill in the rest yourself if you find this helpful.
    • Thread Starter

    Thanks you guys, I now need to write this down in the most concise way I can
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Updated: March 24, 2011
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