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Embeddings and Immersions (Translations of Mathematical Monographs)
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Seller Inventory Satisfaction Guaranteed! Book is in Used-Good condition. Pages and cover are clean and intact. Used items may not include supplementary materials such as CDs or access codes. May show signs of minor shelf wear and contain limited notes and highlighting. Items related to Embeddings and Immersions Translations of Mathematical Embeddings and Immersions Translations of Mathematical Monographs. Masahisa Adachi. Coutinho , p. In the paper in question, Ore considered the generalisation of the theory of determinants to skew fields, with the goal of solving systems of linear equations with coefficients from these.
This was a problem that had seen much study since the mid- s, but Ore was critical of some of the earlier efforts of other mathematicians. It was not his purpose to treat any embedding problem, but one such problem emerged along the way.
For the non-commutative case v. Ore, , p. However, it quickly emerged that this notion was inadequate for his purposes, since it did not allow for all the operations needed when solving systems of linear equations: We shall in the following consider systems of linear equations with coefficients which are elements of such a ring.
In order to perform an elimination to obtain a solution of a linear system, it seems necessary that the coefficients should satisfy the axioms mentioned The main operation for the usual elimination is however to multiply one equation by a factor and another equation by another factor to make the coefficients of one of the unknowns equal in the two equations. Existence of common multiplum.
It is easy to see that this new condition makes possible the operation indicated above. To any non-commutative ring which satisfies MV and which has no zero divisors, Ore gave the name regular ring;10 he did not assume the existence of a multiplicative identity. What Ore called a regular ring with identity is now termed a right Ore domain Coutinho, , p. Embedding semigroups in groups 9 Thus, any non-commutative ring without zero divisors, and in which any two ele- ments have a common right multiple, can be embedded in a skew field.
Complications not present in the commutative case emerge in the course of the proof, but condition MV serves to resolve these.
The first point that it was necessary for him to address was that of equality of fractions, that is, the derivation of a non-commutative version of condition 1. The proof that the notion of equality defined in 5 is transitive needs special care, but is again an easy exercise see Ore , p. I include it here for comparison with the corresponding proof in the commutative case see Section 2.
Condition MV is thus shown to be sufficient for the embedding of a non-commutative ring in a skew field. In particular, he aimed to find those rings in which it is possible to define such a determinant, and in which we may therefore solve systems of linear equations by elimination. Regular rings are precisely the rings that Ore sought. In the introduction to the paper, he commented In the final section of the paper, Ore extended these ideas to determinants of nth order. Moreover, the methods employed by Steinitz and Ore were almost entirely multiplicative in nature, and may therefore be adapted immediately to the semigroup case.
In this way, we obtain two theorems concerning the embedding of a commutative cancellative semigroup, and a particular non-commutative cancellative semigroup, in a group, namely, its group of fractions: the multiplicative group of the skew field of fractions obtained via the construction of Steinitz or Ore. As noted in the introduction, however, this adaptation to the semigroup case seems to have been regarded as so obvious that it was a long time before any mathematician took the trouble to write down the theorems rigorously.
The earliest explicit statement and proof of these results came in the work of Dubreil, which we will see in Section 4. Embedding semigroups in groups 11 3 Early results for semigroups 3. Sush- kevich. To modern eyes, this decom- position is somewhat simple-minded, for it is merely the separation of a semigroup into its group of units and the two-sided ideal formed of all non-units. Using this terminology, he had been able to make some elementary observations, such as the fact that the group part of a semigroup is non-empty if and only if the semigroup has an identity element.
In particular, the principal part of a given cancellative semi- group, as a cancellative semigroup itself, was necessarily infinite; there was no such restriction on the group part. In the paper of , Sushkevich began with a cancellative semigroup S; the group part of S was denoted by G, and its principal part by H. To this end, he considered separately the case where S has an identity element, and that where it does not.
I give here a sketch of his method in each instance: 12 A two-sided ideal in a semigroup is defined in the same way as for a ring, but without any mention of addition; see Hollings a, p. The other author was Fritz Klein—Barmen Thus, e is a two-sided identity for S. It is clear that H forms a cancellative semigroup, anti-isomorphic to H. Next, introduce another new element E and define it to be a two-sided identity for all elements of both H and H. Two alternating products are deemed equivalent if one may be obtained from the other by the application of the rules for multiplication in H or H, or through the insertion or deletion of factors of the form XX or XX; otherwise, two such products are distinct.
He argued briefly that H1 is in fact a group. The semigroup H is thus, apparently, embedded in the group H1. The only products that are not already defined within this union are those of elements from H with elements from G. It follows that G is contained in H1. At this point, Sushkevich had thus, apparently, embedded an arbitrary cancellative semigroup in a group — indeed, the same group as in case 1. We know from the comments in the introduction and will see in more detail in the next subsection that Sushkevich must have gone wrong somewhere — indeed, the error lies in his definition of H1 , as we will see below.
However, since this second method also employs the above con- struction for H1 , we need not go into it here. This is perhaps because, as we will see in the next subsection, the error was picked up very quickly. The relevant passages of the monograph are, however, somewhat muddled, and appear to contain a needless to say, unsuccessful attempt to patch up the proof of the embedding theorem. He had graduated with a first degree from Moscow State University in but retained a Moscow University affiliation in connection with the research work that he was engaged in by the mids see Section 5.
Nikolskii , English translation, p. In neither case, however, did he mention the semigroup case. I make this assertion because of the style in which Maltsev subsequently cited Sushkevich in his own paper: rather than us- ing Ukrainian surely an easy option for a native Russian speaker? I contend that Maltsev would not have made this error himself, had he seen the paper.
However, the analogous question concerning non-commutative semigroups, as far as I know, remained unsolved. Suschkewitsch has published a proof However, we shall construct Malcev, , p. Maltsev attached a footnote to the end of this sentence, explaining that the result may be proved in much the same way as for integral domains: via the construction of the group of fractions.
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We thus have, seemingly for the first time, an explicit statement of Theorem 4, if not an explicit proof. It is therefore not unreasonable to suggest that the inspiration for his work came from a reading of van der Waerden.
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The parenthetic German term in the above quotation as well as in one below is also quite suggestive. After stating the embedding problem for cancellative semigroups, he commented: An analogous problem exists for rings, viz. Since the multiplicative semigroup of this ring contains H, it cannot be embedded in a skew field. We may therefore speak of inverses of elements of H within the group.
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Further rearrangement yields In general, however, condition Z is not sufficient see, for example, the later counter- example of Holvoet Nevertheless, with the necessity of condition Z estab- lished, Maltsev was able to outline his procedure for the construction of the required counterexample: Hence [it] follows that if a [cancellative] semigroup H does not satisfy the condition Z then this semigroup can not be immersed into a group.
In the next [section] we shall construct a [cancellative] semigroup not satisfying the condition Z. It follows that H may not be embeddedPin a group. Maltsev thus constructed a non-commutative ring without zero divi- sors which is not embeddable in a skew field, and whose multiplicative semigroup is not embeddable in a group.
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This later give rise to a more subtle problem: are there rings which do not embed in skew fields but whose multiplicative semigroups embed in groups? Indeed, it appears that even as early as 12th April the date at the end of the paper , Maltsev had already obtained the deeper results that we will consider in Section 5, for we find the following in his introduction: We have also found the necessary and sufficient conditions for the possibility of immersion of a semigroup into a group.
However these are too complicated to be included in this paper. Following his material on embeddings in Theory of generalised groups, Sushkevich does not appear to have attempted to make any further contributions in this area. During this conference, both Sushkevich and Maltsev delivered lectures in the afternoon session of 16th Novem- ber, one after the other. Before 19 F. R, op. In this paper, Dubreil considered not merely the possibility of embedding a semigroup in a group, but also the manner in which the embedding may be realised.
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Clifford and G. Clifford and Preston, , p. Richter, to whom I express my sincere thanks. Thus, although right regularity is only sufficient for embedding in general, it is both necessary and sufficient for embedding specifically into a group of left fractions.