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A ring $A$ is called real iff the following identity holds for all $n\in \N$ : $$a_1^2 + \dots + a_n^2 = 0 \Leftrightarrow a_1,\dots,a_n = 0 \qquad (\forall a_1,\dots,a_n\in A)$$
Remark 1 If $A$ is a ring then being real implies the following
Conversely, we note that if $A$ is reduced and can have a partial ordering then $A$ is a real ring. If $A$ is a field then we call it a real field. Similarly we define real domains, real (von Neumann) regular rings, $\dots$
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"real ring" is owned by jocaps.
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Cross-references: regular rings, domains, field, conversely, reduced, partial ordering, implies, identity, iff, real, ring
This is version 3 of real ring, born on 2009-03-20, modified 2009-03-21.
Object id is 11673, canonical name is RealRing.
Accessed 310 times total.
Classification:
| AMS MSC: | 13J25 (Commutative rings and algebras :: Topological rings and modules :: Ordered rings) | | | 13J30 (Commutative rings and algebras :: Topological rings and modules :: Real algebra) |
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Pending Errata and Addenda
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