# 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 divisor of $R$.

Examples

1. 1.

The ring $\mathbb{Q}$ and its subring $\mathbb{Z}$ have the common unity 1.

2. 2.

The subring $S$ of even integers of the ring $\mathbb{Z}$ has no unity.

3. 3.

Let $S$ be the subring of all pairs $(a,\,0)$ of the ring  $R=\mathbb{Z}\times\mathbb{Z}$  for which the operations$+$” and “$\cdot$” 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  $R=\{(a,\,2b)|\,\,\,a\in\mathbb{Z}\,\land\,b\in\mathbb{Z}\}$ (operations componentwise).  Now $S$ has the unity $(1,\,0)$ but $R$ has no unity.

5. 5.

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

Title unities of ring and subring UnitiesOfRingAndSubring 2013-03-22 14:49:37 2013-03-22 14:49:37 pahio (2872) pahio (2872) 5 pahio (2872) Result msc 13-00 msc 16-00 msc 20-00 UnityOfSubring