Results 1 to 9 of 9

Math Help - Subrings

  1. #1
    Senior Member
    Joined
    Apr 2008
    From
    Vermont
    Posts
    318

    Subrings

    Which of the following are sets are subrings of the field R of real numbers?
    a) A={m+n(2)^(1/2)|m,n in the integers and n is even}
    b)B={m+n(2)^(1/2)|m,n in the integers and m is odd}
    c)C={a+b(2)^(1/3)|a,b in the rationals}
    d) D={a+b(3)^(1/3)+c(9)^(1/3)|a,b, c in the rationals}

    My problem is getting started. I know a set R is a subring if it is closed under addition and multiplication, if a is in R, the -a is in R, R contains the identity
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    Just check the things you mentioned. For example, B is clearly not closed under addition. Also, it is not necessarily required that a subring contain 1, although it's possible that in your setting, rings are assumed to have multiplicative identities.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Apr 2008
    From
    Vermont
    Posts
    318
    ok so m+n(2)^1/2+x+y(2)^1/2, m is even so call it 2b
    2b+n(2)^1/2+2a+y(2)^1/2
    2(a+b)+(n+y)2^1/2 Ok that makes sense for checking closure of addition. I think I can figure out multiplication. Now what about the a and -a stuff?
    a is in m+n(2)^1/2. Now I get stuck on going further.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Feb 2008
    Posts
    410
    Well it looks like you're trying to show that A is closed under addition, but if so then you have misread the set. We need n even for m+n\sqrt{2}. However, your method is more or less correct once you flip those back around. We have (m_1+n_1\sqrt{2})+(m_2+n_2\sqrt{2})=(m_1+m_2)+(n_1  +n_2)\sqrt{2}, and this preserves the conditions of A. So A is closed under addition. Also notice that if m+n\sqrt{2}\in A, then -m-n\sqrt{2} is the additive inverse, and this is also an element of A. So A is closed under additive inverses. Note also that 1\in A. So you just need to show that A is closed under multiplication.

    Let m_1+n_1\sqrt{2} and m_2+n_2\sqrt{2} be members of A. If you multiply them, do you get an element of A? If so, then A is a subring. Otherwise it is not.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Apr 2008
    From
    Vermont
    Posts
    318
    Ok I follow all that except for why 1 is in A. Is that because if we let m=1 and n=0?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Feb 2008
    Posts
    410
    Quote Originally Posted by kathrynmath View Post
    Ok I follow all that except for why 1 is in A. Is that because if we let m=1 and n=0?
    Yes, that is correct. n=0 is even and so 1\in A.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Apr 2008
    From
    Vermont
    Posts
    318
    D isn't a subring, right?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Feb 2008
    Posts
    410
    Why isn't D a subring?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    Apr 2008
    From
    Vermont
    Posts
    318
    because it's not closed under multiplication. I got that it was closed under addition, but not closed under multiplication
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subrings and subgroups of Z
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 3rd 2011, 11:47 PM
  2. Homomorphism and Subrings
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 28th 2010, 04:15 PM
  3. subfield, subrings
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 11th 2010, 07:11 PM
  4. subrings
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: December 8th 2008, 05:12 PM
  5. Algebra....subrings
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 28th 2006, 06:23 PM

Search Tags


/mathhelpforum @mathhelpforum