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# unitization

The operation of unitization allows one to add a unity element to an algebra. Because of this construction, one can regard any algebra as a subalgebra of an algebra with unity. If the algebra already has a unity, the operation creates a larger algebra in which the old unity is no longer the unity.

Let $\bf A$ be an algebra over a ring $\bf R$ with unity $1$. Then, as a module, the unitization of $\bf A$ is the direct sum of $\bf R$ and $\bf A$:

$\bf A^{+}=\bf R\oplus\bf A$ |

The product operation is defined as follows:

$(x,a)\cdot(y,b)=(xy,ab+xb+ya)$ |

The unity of $\bf A^{+}$ is $(1,0)$.

It is also possible to unitize any ring using this construction if one regards the ring as an algebra over the ring of integers. (See the entry every ring is an integer algebra for details.) It is worth noting, however, that the result of unitizing a ring this way will always be a ring whose unity has zero characteristic. If one has a ring of finite characteristic $k$, one can instead regard it as an algebra over $\mathbb{Z}_{k}$ and unitize accordingly to obtain a ring of characteristic $k$.

The construction described above is often called “minimal unitization”. It is in fact minimal, in the sense that every other unitization contains this unitization as a subalgebra.

## Mathematics Subject Classification

16-00*no label found*13-00

*no label found*20-00

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