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# 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 non-zero unities but these are distinct.

4. $R$ has no unity but $S$ has a non-zero 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 $(a,\,0)$ 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. 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. Neither the ring $\{(2a,\,2b)|\,\,\,a,\,b\in\mathbb{Z}\}$ (operations componentwise) nor its subring consisting of the pairs $(2a,\,0)$ have unity.

## Mathematics Subject Classification

13-00*no label found*16-00

*no label found*20-00

*no label found*

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