Imbedding of rings

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 [1], 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. [2]). 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 [2]. The class of associative rings imbeddable in skew-fields is not finitely axiomatized (i.e. cannot be defined by a finite number of axioms) [3]. 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 [4]. The group algebra of an ordered group is imbeddable in a skew-field (the Mal'tsev–Neumann theorem, cf. [4]). An arbitrary domain of free right (left) ideals (cf. Associative rings and algebras) is imbeddable in a skew-field [2].

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 [4].

References

[1]	A.I. [A.I. Mal'tsev] Malcev, "On the immersion of an algebraic ring into a field" Math. Ann. , 113 (1937) pp. 686–691
[2]	P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
[3]	P.M. Cohn, "The class of rings embeddable in skew fields" Bull. London Math. Soc. , 6 (1974) pp. 147–148
[4]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[5]	L.A. Bokut', "Embedding of rings" Russian Math. Surveys , 42 : 4 (1987) pp. 105–138 Uspekhi Mat. Nauk , 42 (1987) pp. 87–111

Comments

A $T$-homomorphism is also called a localization (cf. also Localization in a commutative algebra).

Another classical problem is imbedding of a ring $R$ in a finite matrix ring over a commutative ring. A necessary condition is that it satisfies all universal polynomial identities $p[x_1,\ldots,x_m]=0$ of the $n\times n$ matrix ring over the integers. The condition is sufficient if $R$ is prime or semi-prime, but fails in other cases (cf. [a1]).

References

[a1]	L.H. Rowen, "Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7