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center (rings) (Definition)

If $ A$ is a ring, the center of $ A$, sometimes denoted $ \operatorname{Z}(A)$, is the set of all elements in $ A$ that commute with all other elements of $ A$. That is,

$\displaystyle \operatorname{Z}(A) = \{ a \in A \mid ax = xa$   $\displaystyle \forall x \in A \}$

Note that $ 0 \in \operatorname{Z}(A)$ so the center is non-empty. If we assume that $ A$ is a ring with a multiplicative unity $ 1$, then $ 1$ is in the center as well. The center of $ A$ is also a subring of $ A$.



"center (rings)" is owned by drini. [ owner history (2) ]
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See Also: center of a group

Other names:  center
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Cross-references: subring, unity, multiplicative, ring
There are 7 references to this entry.

This is version 3 of center (rings), born on 2002-06-07, modified 2002-06-09.
Object id is 3065, canonical name is CenterOfARing.
Accessed 4497 times total.

Classification:
AMS MSC: 16U70 (Associative rings and algebras :: Conditions on elements :: Center, normalizer )
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