unities of ring and subring


Let R be a ring and S a proper subring of it.  Then there exists five cases in all concerning the possible unities of R and S.

  1. 1.

    R and S have a common unity.

  2. 2.

    R has a unity but S does not.

  3. 3.

    R and S both have their own non-zero unities but these are distinct.

  4. 4.

    R has no unity but S has a non-zero unity.

  5. 5.

    Neither R nor S have unity.

Note:   In the cases 3 and 4, the unity of the subring S must be a zero divisorMathworldPlanetmath of R.

Examples

  1. 1.

    The ring and its subring have the common unity 1.

  2. 2.

    The subring S of even integers of the ring has no unity.

  3. 3.

    Let S be the subring of all pairs (a, 0) of the ring  R=×  for which the operationsMathworldPlanetmath+” and “” are defined componentwise (i.e.  (a,b)+(c,d)=(a+c,b+d)  etc.).  Then S and R have the unities (1, 0) and (1, 1), respectively.

  4. 4.

    Let S be the subring of all pairs (a, 0) of the ring  R={(a, 2b)|ab} (operations componentwise).  Now S has the unity (1, 0) but R has no unity.

  5. 5.

    Neither the ring {(2a, 2b)|a,b} (operations componentwise) nor its subring consisting of the pairs (2a, 0) have unity.

Title unities of ring and subring
Canonical name UnitiesOfRingAndSubring
Date of creation 2013-03-22 14:49:37
Last modified on 2013-03-22 14:49:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Result
Classification msc 13-00
Classification msc 16-00
Classification msc 20-00
Related topic UnityOfSubring