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.
$R$ and $S$ have a common unity.

2.
$R$ has a unity but $S$ does not.

3.
$R$ and $S$ both have their own nonzero unities but these are distinct.

4.
$R$ has no unity but $S$ has a nonzero unity.

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

2.
The subring $S$ of even integers of the ring $\mathbb{Z}$ has no unity.

3.
Let $S$ be the subring of all pairs $(a,\mathrm{\hspace{0.17em}0})$ 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 $(1,\mathrm{\hspace{0.17em}0})$ and $(1,\mathrm{\hspace{0.17em}1})$, respectively.

4.
Let $S$ be the subring of all pairs $(a,\mathrm{\hspace{0.17em}0})$ of the ring $R=\{(a,\mathrm{\hspace{0.17em}2}b)a\in \mathbb{Z}\wedge b\in \mathbb{Z}\}$ (operations componentwise). Now $S$ has the unity $(1,\mathrm{\hspace{0.17em}0})$ but $R$ has no unity.

5.
Neither the ring $\{(2a,\mathrm{\hspace{0.17em}2}b)a,b\in \mathbb{Z}\}$ (operations componentwise) nor its subring consisting of the pairs $(2a,\mathrm{\hspace{0.17em}0})$ have unity.
Title  unities of ring and subring 

Canonical name  UnitiesOfRingAndSubring 
Date of creation  20130322 14:49:37 
Last modified on  20130322 14:49:37 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  5 
Author  pahio (2872) 
Entry type  Result 
Classification  msc 1300 
Classification  msc 1600 
Classification  msc 2000 
Related topic  UnityOfSubring 