Finite and infinite primes for rings and fields

Harrison, David K.

doi:10.1090/memo/0068

Cited by 47 publications

(70 citation statements)

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“…For some examples, see [1,Theorem 1], [3,Proposition 7], and [6, p. 317]. By introducing similar concepts in abelian groups, C. W. Kohls [5] has given simpler more direct proofs of Harrison's and Krivine's results as well as obtaining a representation theorem for groups.…”

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confidence: 99%

“…It was hoped that [3,Proposition 7] could be extended so as to obtain representation of a certain class of rings as subrings of the division ring of real quaternions, for it is noted that many subrings of the quaternions satisfy our hypotheses. What we actually obtain is representation of such rings as subrings of division algebras that are algebraic over subfields of the reals.…”

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confidence: 99%

“…Rings without identity. We here extend [1,Theorem 1] and [3,Proposition 7] to rings without identity.…”

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confidence: 99%

“…Therefore, 1 is an 77'-unit in A'. The existence of the desired monomorphism thus follows from [3,Proposition 7].…”

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Embedding a Partially Ordered Ring in a Division Algebra

Reynolds¹

1971

Transactions of the American Mathematical Society

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Abstract.D. K. Harrison has shown that if a ring with identity has a positive cone that is an infinite prime (a subsemiring that contains 1 and is maximal with respect to avoiding -1), and if the cone satisfies a certain archimedean condition for all elements of the ring, then there exists an order isomorphism of the ring into the real field. Our main result shows that if Harrison's archimedean condition is weakened so as to apply only to the elements of the cone and if a certain centrality relation is satisfied, then there exists an order isomorphism of the ring into a division algebra that is algebraic over a subfield of the real field. Also, Harrison's result and a related theorem of D. W. Dubois are extended to rings without identity; in so doing, it is shown that order isomorphic subrings of the real field are identical. It was hoped that [3, Proposition 7] could be extended so as to obtain representation of a certain class of rings as subrings of the division ring of real quaternions, for it is noted that many subrings of the quaternions satisfy our hypotheses. What we actually obtain is representation of such rings as subrings of division algebras that are algebraic over subfields of the reals. However, if the rings are commutative, the desired result follows; that is, they are then isomorphic to subrings of the complex field. In the general case, we can show that the semiring of nonnegative reals satisfies our hypotheses as a subsemiring of a ring A iff A is isomorphic to either the real field, the complex field, or the quaternions. This is quite analogous to Frobenius' theorem [8] which states that an algebraic real division algebra is isomorphic to one of these; indeed we use his theorem in our proof.

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Embedding a Partially Ordered Ring in a Division Algebra

Reynolds¹

1971

Transactions of the American Mathematical Society

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“…A conic preprime is simply a positive cone and induces a partial order: x^y<=>y^x<=>x -yeP. A preprime P is Archimedean if for all x in K there exists a natural number n with n -x in P, (condition (2) in the definition of Stone ring) and is (AC) if from 1 + nxe P for all ne Nfollows xe P (condition (3)). We redefine a Stone ring as a pair ζK, P> where P is an infinite conic Archimedean (AC) preprime in if.…”

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confidence: 99%