Results 1 to 7 of 7

Math Help - order of ring and idempotent

  1. #1
    Newbie
    Joined
    May 2008
    Posts
    12

    order of ring and idempotent

    Q1) Consider the matrix ring M2(Z2)
    a) Find the order of the ring
    b) List all the units of the ring

    Q3) an element a in a ring is an idempotent if a^2=a
    show the set of idempotents in a commutative ring are closed under multiplication and find the idempotents in
    Z8 X Z12
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by Ryan0710 View Post
    Q1) Consider the matrix ring M2(Z2)
    a) Find the order of the ring
    b) List all the units of the ring
    a) EDIT: I was wrong. The order as seen below is 8. I apologise for my mistake. Hey Ryan, how come you accepted my answer when it was not convincing?

    b)For the multiplicative inverse to exist , the determinant must be non-zero. Now since the determinant for \left(\begin{matrix} a & b \\ c & d \end{matrix}\right) is ad-bc. And since a,b,c,d \in \mathbb{Z}_2, either ad = 1 or bc = 1 is the only possible solution.

    Thus the only units are:

    With ad = 1,

    \left(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right),\left(\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}\right),\left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right)<br />

    With bc =1,
    \left(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right),\left(\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}\right),\left(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\right)
    Last edited by Isomorphism; June 13th 2008 at 11:21 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by Ryan0710 View Post
    Q3) an element a in a ring is an idempotent if a^2=a
    show the set of idempotents in a commutative ring are closed under multiplication and find the idempotents in
    Z8 X Z12
    let E \subset R and E is the set of idempotent elements of R.

    if x,y \in E then x^2,y^2 \in E

    x\cdot y=x^2\cdot y^2=(x\cdot x)\cdot(y \cdot y)

    Using both the associtative and communitive properties we get

    =(x\cdot y)\cdot(x \cdot y)=(xy)\cdot(xy)=(xy)^2 \in E

    Therefore E is closed


    Edit:Sorry about the mistake. Thanks NonCommAlg.
    Last edited by TheEmptySet; June 13th 2008 at 09:33 PM. Reason: typo, okay three of them :( and I'm incorrect.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    2 little points:

    1) the order of a ring is the number of elements of that ring. thus the order of M_2(\mathbb{Z}_2) is 8.

    2) in \mathbb{Z}_8 only 0 and 1 are idempotent. the idempotents of \mathbb{Z}_{12} are 0, 1, 4 and 9. so there are

    8 idempotents in \mathbb{Z}_8 \times \mathbb{Z}_{12}. what are they Ryan0710 ?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by NonCommAlg View Post
    2 little points:

    1) the order of a ring is the number of elements of that ring. thus the order of M_2(\mathbb{Z}_2) is 8.
    I am sorry, I am not a student of algebra in a formal school. I just read this. It claimed the order of the ring is the order of the additive group. Isnt the order of the additive group 2?

    To Ryan,

    I must be wrong, NonCommAlg is a good algebraist, there is no way he would have been wrong on such "elementary" things

    I apologise for the mistake.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Isomorphism View Post
    I am sorry, I am not a student of algebra in a formal school. I just read this. It claimed the order of the ring is the order of the additive group. Isnt the order of the additive group 2?

    To Ryan,

    I must be wrong, NonCommAlg is a good algebraist, there is no way he would have been wrong on such "elementary" things

    I apologise for the mistake.
    the order of a (finite) group is the number of elements of that group. by the additive group of a ring R, we mean the ring R itself

    considered as an (abelian) additive group, i.e. (R, +). so the set is still the ring R, but considered with + only. so the order of a (finite)

    ring is the number of elements of that ring. regarding your question, 2 is the order of every (non-zero) element of M_2(\mathbb{Z}_2), and not

    the order of the ring.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by NonCommAlg View Post
    the order of a (finite) group is the number of elements of that group. by the additive group of a ring R, we mean the ring R itself

    considered as an (abelian) additive group, i.e. (R, +). so the set is still the ring R, but considered with + only. so the order of a (finite)

    ring is the number of elements of that ring. regarding your question, 2 is the order of every (non-zero) element of M_2(\mathbb{Z}_2), and not

    the order of the ring.
    Oh!

    I am sorry. I confused the order of the group with the order of an element.

    Thank you
    Last edited by ThePerfectHacker; June 14th 2008 at 06:54 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: November 28th 2009, 02:07 AM
  2. Idempotent Matrix
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 20th 2009, 01:02 PM
  3. order of an ideal ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 19th 2009, 08:12 PM
  4. Idempotent linear map
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: January 20th 2009, 10:49 PM
  5. Idempotent Matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 3rd 2008, 04:05 PM

Search Tags


/mathhelpforum @mathhelpforum