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A [[Monomorphism|monomorphism]] of a ring into another ring; a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502101.png" /> is imbeddable in a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502102.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502103.png" /> is isomorphic to a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502104.png" />. The conditions for imbedding of an associative ring in an (associative) [[Skew-field|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 [[#References|[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. [[#References|[2]]]). A square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502105.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502106.png" /> over an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502107.png" /> is said to be non-full if it can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502108.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i0502109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021010.png" /> are matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021012.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021013.png" />. Let
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{{TEX|done}}
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A [[Monomorphism|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|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 [[#References|[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. [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021014.png" /></td> </tr></table>
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$$A=(a,a_2,\ldots,a_n),\quad B=(b,a_2,\ldots,a_n)$$
  
be square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021015.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021016.png" /> in which all columns (except, possibly, the first column) are identical. Then the matrix
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be square matrices of order $n\times n$ over $R$ in which all columns (except, possibly, the first column) are identical. Then the matrix
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021017.png" /></td> </tr></table>
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$$C=(a+b,a_2,\ldots,a_n)$$
  
is said to be the determinant sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021020.png" /> with a unit element is imbeddable in a skew-field if and only if it has no zero divisors and if no scalar matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021021.png" /> with a non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021022.png" /> along the diagonal can be represented as the determinant sum of a finite number of non-full matrices [[#References|[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) [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021023.png" /> 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|Imbedding of semi-groups]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021024.png" /> is imbeddable in a skew-field [[#References|[4]]]. The group algebra of an ordered group is imbeddable in a skew-field (the Mal'tsev–Neumann theorem, cf. [[#References|[4]]]). An arbitrary domain of free right (left) ideals (cf. [[Associative rings and algebras|Associative rings and algebras]]) is imbeddable in a skew-field [[#References|[2]]].
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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 [[#References|[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) [[#References|[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|Imbedding of semi-groups]]). Then $R$ is imbeddable in a skew-field [[#References|[4]]]. The group algebra of an ordered group is imbeddable in a skew-field (the Mal'tsev–Neumann theorem, cf. [[#References|[4]]]). An arbitrary domain of free right (left) ideals (cf. [[Associative rings and algebras|Associative rings and algebras]]) is imbeddable in a skew-field [[#References|[2]]].
  
A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021025.png" /> is imbeddable in a division ring if and only if it has no zero divisors. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021027.png" /> be rings, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021028.png" /> be a symbol, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021029.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021030.png" /> is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021032.png" />-homomorphism if: 1) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021033.png" /> is a ring and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021034.png" /> on this set is a ring homomorphism; 2) it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021036.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021037.png" />; and 3) it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021039.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021040.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021041.png" />-homomorphism of a field is nothing but a specialization (of a point) of the field (cf. [[Specialization of a point|Specialization of a point]]). A division ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021042.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021044.png" />-extension of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021045.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021046.png" /> includes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021047.png" /> and is generated (as a division ring) by the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021048.png" />, while any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021049.png" />-homomorphism of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021050.png" /> into some division ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021051.png" /> may be extended to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021052.png" />-homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021053.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021054.png" />. Every ring without zero divisors has a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021055.png" />-extension [[#References|[4]]].
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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|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 [[#References|[4]]].
  
 
====References====
 
====References====
Line 17: Line 18:
  
 
====Comments====
 
====Comments====
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021056.png" />-homomorphism is also called a localization (cf. also [[Localization in a commutative algebra|Localization in a commutative algebra]]).
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A $T$-homomorphism is also called a localization (cf. also [[Localization in a commutative algebra|Localization in a commutative algebra]]).
  
Another classical problem is imbedding of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021057.png" /> in a finite matrix ring over a commutative ring. A necessary condition is that it satisfies all universal polynomial identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021058.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021059.png" /> matrix ring over the integers. The condition is sufficient if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050210/i05021060.png" /> is prime or semi-prime, but fails in other cases (cf. [[#References|[a1]]]).
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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. [[#References|[a1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Polynomial identities in ring theory" , Acad. Press  (1980)  pp. Chapt. 7</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Polynomial identities in ring theory" , Acad. Press  (1980)  pp. Chapt. 7</TD></TR></table>

Latest revision as of 14:57, 30 July 2014

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
How to Cite This Entry:
Imbedding of rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_of_rings&oldid=19119
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article