# Imbedding of rings

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A monomorphism of a ring into another ring; a ring $R$ is imbeddable in a ring $L$ if $R$ is isomorphic to a subring of $L$. The conditions for imbedding of an associative ring in an (associative) skew-field and of an arbitrary ring into a division ring have been studied in great detail. These studies were initiated by A.I. Mal'tsev , who constructed an example of an associative ring without zero divisors and not imbeddable in a skew-field. The following Mal'tsev problem remained an open question for a long time: Is any associative ring without zero divisors and for which the semi-group of non-zero elements is imbeddable in a group, imbeddable in a skew-field? This problem was negatively answered in 1966 (cf. ). A square matrix $A$ of order $n\times n$ over an associative ring $R$ is said to be non-full if it can be represented in the form $A=BC$ where $B$, $C$ are matrices of orders $n\times r$ and $r\times n$, respectively, and $r<n$. Let

$$A=(a,a_2,\ldots,a_n),\quad B=(b,a_2,\ldots,a_n)$$

be square matrices of order $n\times n$ over $R$ in which all columns (except, possibly, the first column) are identical. Then the matrix

$$C=(a+b,a_2,\ldots,a_n)$$

is said to be the determinant sum of $A$ and $B$ with respect to the first column. The determinant sum of square matrices of the same order with respect to an arbitrary column (row) is defined in a similar manner. An associative ring $B$ with a unit element is imbeddable in a skew-field if and only if it has no zero divisors and if no scalar matrix $aE$ with a non-zero element $a$ along the diagonal can be represented as the determinant sum of a finite number of non-full matrices . The class of associative rings imbeddable in skew-fields is not finitely axiomatized (i.e. cannot be defined by a finite number of axioms) . A number of sufficient conditions for imbedding of an associative ring in a skew-field are known; the following are the most important. Let $R$ be an associative ring without zero divisors and for which the semi-group of non-zero elements satisfies Ore's condition (cf. Imbedding of semi-groups). Then $R$ is imbeddable in a skew-field . The group algebra of an ordered group is imbeddable in a skew-field (the Mal'tsev–Neumann theorem, cf. ). An arbitrary domain of free right (left) ideals (cf. Associative rings and algebras) is imbeddable in a skew-field .

A ring $R$ is imbeddable in a division ring if and only if it has no zero divisors. Let $R$, $L$ be rings, let $\infty$ be a symbol, $\infty\not\in L$. A mapping $\phi:R\to\{L,\infty\}$ is said to be a $T$-homomorphism if: 1) the set $\phi^{-1}(L)$ is a ring and the mapping $\phi$ on this set is a ring homomorphism; 2) it follows from $\phi(ab)\neq\infty$, $\phi(a)=\infty$ that $\phi(b)=0$; and 3) it follows from $\phi(ab)\neq\infty$, $\phi(b)=\infty$ that $\phi(a)=0$. A $T$-homomorphism of a field is nothing but a specialization (of a point) of the field (cf. Specialization of a point). A division ring $L$ is a free $T$-extension of a ring $R$ if $L$ includes $R$ and is generated (as a division ring) by the ring $R$, while any $T$-homomorphism of the ring $R$ into some division ring $S$ may be extended to a $T$-homomorphism of $L$ into $S$. Every ring without zero divisors has a free $T$-extension .

How to Cite This Entry:
Imbedding of rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_of_rings&oldid=32568
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article