V. P. ELIZAROV

THE RING OF QUOTIENTS WITH RESPECT TO A PRIME IDEAL

(Presented by Academician A. I. Mal’tsev on 6 XI 1959)

Let \(R\) be an arbitrary associative ring and let \(S\) be some multiplicatively closed system of its elements, possibly containing zero divisors of the ring \(R\), but not zero. If the ring \(R\) has a generalized left ring of quotients with respect to the system \(S\), then by \(I\) we shall denote the two-sided ideal of the ring \(R\) that is the intersection of all its \(S\)-prime ideals, and by \(\varphi\) the homomorphic mapping of the ring \(R\) into the ring \(R_{(S)}\) with kernel \(I\) \((^{1})\).

Lemma 1. For any finite number of elements \(x_i \in R_{(S)}\) there exists an element \(\bar{s}\in S\) such that \(\varphi(\bar{s})x_i=\varphi(\bar{r}_i)\), where \(\bar{r}_i\in R\).

Proof. \(x_i=[\varphi(s_i)]^{-1}(r_i)\), where \(s_i\in S,\ r_i\in R,\ i=1,\ldots,n\). For \(s_1\) and any \(s_0\in S\), by the definition of the ideal \(I\), there are elements \(s'\in S\) and \(r'\in R\) such that \(r's_1-s's_0\in I\), i.e. \(\varphi(r')\varphi(s_1)=\varphi(s')\varphi(s_0)=\varphi(\bar{s})\), where \(\bar{s}=s's_0\in S\). Suppose that for \(i=1,\ldots,n-1\) there exists an element \(\bar{s}'\in S\) such that \(\varphi(\bar{s}')=\varphi(\bar{r}'_i)\varphi(s_i)\), where \(\bar{r}'_i\in R\). For \(\bar{s}'\) and \(s_n\) there are elements \(s''\in S\) and \(r''\in R\) such that \(\varphi(r'')\varphi(s_n)=\varphi(s'')\varphi(\bar{s}')\). Then \(\varphi(\bar{s})=\varphi(s'')\varphi(\bar{s}')=\varphi(\bar{r}_i)\varphi(s_i)\), where \(\bar{r}_i=s''\bar{r}'_i,\ i=1,\ldots,n-1\). Thus \(\varphi(\bar{s})=\varphi(\bar{r}_i)\varphi(s_i)\), where \(\bar{r}_n=r'',\ i=1,\ldots,n\). Then \(\varphi(\bar{r}_i)\varphi(s_i)x_i=\varphi(\bar{r}_i)\varphi(r_i)\), i.e. \(\varphi(\bar{s})x_i=\varphi(\bar{r}'_i)\), where \(\bar{r}'_i\in R\). The lemma is proved.

If \(L\) is a left ideal of the ring \(R\), then by

\[ L^e=R_{(S)}\varphi(L) \]

we shall denote the left ideal of the ring \(R_{(S)}\) generated by the set \(\varphi(L)\). If \(L_{(S)}\) is a left ideal of the ring \(R_{(S)}\), then by

\[ L^c_{(S)}=\varphi^{-1}\bigl(L_{(S)}\cap \varphi(R)\bigr) \]

we shall denote the left ideal of the ring \(R\) that is the complete inverse image of \(L_{(S)}\cap\varphi(R)\).

Lemma 2. The ideal \(L^{ec}\) consists of those elements \(b\in R\) for which there exist elements \(s\in S,\ l\in L\) such that \(sb-l\in I\).

Proof. Let \(b\in R,\ b\in L^{ec}=(R_{(S)}\varphi(L)\cap\varphi(R))\). Then \(\varphi(b)\in R_{(S)}\varphi(L)=L^e\), and

\[ \varphi(b)=\sum_{i=1}^{n} r_s^{(i)}\varphi(l_i), \]

where \(r_s^{(i)}\in R_{(S)},\ l_i\in L\). There exists (Lemma 1) an element \(s\in S\) such that \(\varphi(s)r_s^{(i)}=\varphi(r_i),\ r_i\in R\). Then \(\varphi(s)\varphi(b)=\varphi(l)\), where \(l\in L\). Consequently, \(\varphi(sb-l)=0\) and \(sb-l\in I\). If for \(b\in R\) there are elements \(s\in S\) and \(l\in L\) such that \(sb-l\in I\), then \(\varphi(b)=[\varphi(s)]^{-1}\varphi(l)\in R_{(S)}\varphi(L)=L^e\). But then \(b\in\varphi^{-1}(L^e\cap\varphi(R))=L^{ec}\), and the lemma is proved.

Lemma 3. \(L^e\ne R_{(S)}\) if and only if, for any elements \(s\in S,\ l\in L\), \(l-s\notin I\).

Proof. If \(L^e=R_{(S)}\varphi(L)=R_{(S)}\), then \(L^{ec}=\varphi^{-1}(L^e\cap\varphi(R))=R\). By Lemma 2, for any element \(r\in R\) there exist elements \(s\in S\) and \(l\in L\) such that \(sr-l\in I\). If \(r=s'\in S\), then \(s's-l\in I\), where \(ss'\in S\). If \(L^e\ne R_{(S)}\), then \(R_{(S)}\varphi(L)\ne R_{(S)}\). If \(l-s\in I\), then \(\varphi(l)=\varphi(s)\) and

\(R_{(S)}\varphi(L)=L^{e}\supseteq R_{(S)}\varphi(S)\ni 1\), i.e. \(L^{e}=R_{(S)}\), which is impossible. Hence, \(l-s\notin I\). The lemma is proved.

A two-sided ideal \(P\) of a ring \(R\) will be called prime if from \(r_{1}r_{2}\in P\), where \(r_{1}\in R,\ r_{2}\in R\), it follows that \(r_{1}\in P\) or \(r_{2}\in P\). If \(R\) is an arbitrary associative ring and \(P\) is its proper prime ideal, then denote \(S=R-P\). \(S\) is a multiplicatively closed system of elements of the ring \(R\) without zero. If the ring \(R_{(S)}\) exists, then we denote it by \(R_{(P)}\) and call it the generalized left ring of fractions of the ring \(R\) with respect to the prime ideal \(P\).

Theorem 1. If the ring \(R\) has the ring \(R_{(P)}\), then:

1) the elements of the ring \(R_{(P)}\) not belonging to the ideal \(P^{e}\) have two-sided inverses in \(R_{(P)}\);

2) the elements of the ideal \(P^{e}\) are neither left nor right divisors of unity;

3) every proper ideal of the ring \(R_{(P)}\) is contained in the ideal \(P^{e}\).

Proof. Let \(x\in R_{(P)}\), \(x\notin P^{e}\). Then
\(x=[\varphi(s)]^{-1}\varphi(s_{1})\), where \(s,s_{1}\in S\), and
\(x[\varphi(s_{1})]^{-1}\varphi(s)=[\varphi(s_{1})]^{-1}\varphi(s)x=1\). Hence \(x\) has a two-sided inverse.

Let \(y\) be an arbitrary element of \(P^{e}\). If \(y'y=1\), where \(y'\in R_{(P)}\), then \(P^{e}=R_{(P)}\). If \(p\in P,\ s\in S\) and \(p-s\in I\), then \(p-s\in P\), since \(I\subseteq P\) and \(s\in P\), which is impossible. Therefore, by Lemma 3, \(P^{e}\ne R_{(P)}\), and \(y\) is not a right divisor of unity. If \(yy'=1\), where \(y'\in R_{(P)}\), then \(y'\) is a right divisor of unity, i.e. \(y'\notin P^{e}\). Consequently, there is an element \(y''\in R_{(P)}\) such that \(y'y''=y''y'=1\). Then \((y-y'')y'=0\) and \(y=y''\), i.e. \(y'y=1\) and \(y\) is a right divisor of unity, which is impossible. Consequently, \(y\) is not a left divisor of unity.

The third assertion of the theorem follows directly from the second. The theorem is proved.

An associative ring \(R\) will be called local (on the left, on the right, two-sided) if in the ring \(R\) there exists a unique (left, right, two-sided) maximal ideal.

Corollary. The ring \(R_{(P)}\) is local on the left.

A left ideal \(L\) of the ring \(R\) will be called contracted if \(L=L^{ec}\).

Lemma 4. An ideal \(L\) is contracted if and only if from \(sr-l\in I\), where \(s\in S,\ r\in R,\ l\in L\), it follows that \(r\in L\).

Proof. Suppose that from \(sr-l\in I\) it follows that \(r\in L\), and let \(b\in R,\ b\in L^{ec}\). Then (Lemma 2) there exist elements \(s'\in S\) and \(l'\in L\) such that \(s'b-l'\in I\). By assumption, \(b\in L\). Thus \(L^{ec}\subseteq L\). On the other hand, \(L\subseteq L^{ec}\), i.e. \(L=L^{ec}\).

Now suppose that \(L=L^{ec}\) and \(sr-l\in I\). Then \(\varphi(s)\varphi(r)=\varphi(l)\) and
\(\varphi(r)=[\varphi(s)]^{-1}\varphi(l)\in R_{(S)}\varphi(L)=L^{e}\). Hence
\(r\in\varphi^{-1}(L^{e}\cap \varphi(R))=L^{ec}=L\). The lemma is proved.

If \(R,\ R_{1}\) and \(R_{2}\) are associative rings and \(\varphi_{1},\varphi_{2}\) are homomorphic mappings of the ring \(R\) into \(R_{1}\) and \(R_{2}\), respectively, then we shall say that \(R_{1}\cong R_{2}\) over \(R\) if there exists an isomorphism of the rings \(R_{1}\) and \(R_{2}\) under which the images of one and the same element of the ring \(R\) correspond to one another.

If, alongside the left generalized ring of fractions \(R_{(S)}\) of the ring \(R\) with respect to the system \(S\), there exists the right generalized ring of fractions \(\overline{R}_{(S)}\) with respect to the same system \(S\) (it can be defined analogously to the way the ring \(R_{(S)}\) is defined in [1]), then it is easy to verify that \(R_{(S)}\cong \overline{R}_{(S)}\) over \(R\) if and only if the kernel \(I\) of the mapping \(\varphi\) of the ring \(R\) into the ring \(R_{(S)}\) coincides with the kernel \(\overline{I}\) of the mapping \(\overline{\varphi}\) of the ring \(R\) into the ring \(\overline{R}_{(S)}\).

Lemma 5. If the ring \(R\) has the rings \(R_{(S)}\), \(\overline{R}_{(S)}\) and \(R_{(S)}\cong \overline{R}_{(S)}\) over \(R\); \(A\) is a two-sided ideal of the ring \(R\) which is contracted as a right ideal, then the left ideal \(A^{e}\) is two-sided in the ring \(R_{(S)}\).

Proof. \(A^{e}=R_{(S)}\varphi(A)=\{\sum r_s^{(a)}\varphi(a_a)\}\), where \(r_s^{(a)}\in R_{(S)}\), \(a_a\in A\). Let \(a^e\in A^e\) and \(x\in R_{(S)}\). By the hypothesis of the theorem, for any elements \(r\in R\) and \(s\in S\) there exist elements \(r'\in R\) and \(s'\in S\) such that \(rs'-sr'\in \bar I=I\). If \(x=[\varphi(s)]^{-1}\varphi(r)\), then \(\varphi(r)\varphi(s')=\varphi(s)\varphi(r')\) and
\(x\varphi(s')=[\varphi(s)]^{-1}\varphi(r)\varphi(s')=[\varphi(s)]^{-1}\varphi(s)\varphi(r')=\varphi(r')\), i.e. \(x=\varphi(r')[\varphi(s')]^{-1}\). Then

\[ a^e x=[r_s^{(1)}\varphi(a_1)+\cdots+r_s^{(n)}\varphi(a_n)]\varphi(r')[\varphi(s')]^{-1}. \]

For the elements \(r_s^{(i)}\) there is (Lemma 1) an element \(\bar s\in S\) such that \(\varphi(\bar s)r_s^{(i)}=\varphi(\bar r_i)\), where \(\bar r_i\in R\). Then

\[ \varphi(\bar s)a^e x=[\varphi(\bar r_1a_1r')+\cdots+\varphi(\bar r_na_nr')][\varphi(s')]^{-1} =[\varphi(a'_1)+\cdots+\varphi(a'_n)][\varphi(s')]^{-1}, \]

where \(\bar r_ia_ir'=a'_i\in A\). Hence \(\varphi(\bar s)a^ex=\varphi(a)[\varphi(s')]^{-1}\), where \(a\in A\), and

\[ a^ex=[\varphi(\bar s)]^{-1}\varphi(a)[\varphi(s')]^{-1}. \]

But \(\varphi(a)[\varphi(s')]^{-1}\in R_{(S)}\), i.e. \(\varphi(a)[\varphi(s')]^{-1}=[\varphi(s_0)]^{-1}\varphi(r_0)\), \(s_0\in S\), \(r_0\in R\). Then \(\varphi(r_0)\varphi(s')=\varphi(s_0)\varphi(a)=\varphi(a')\), where \(a'\in A\). Consequently, \(r_0s'-a'\in I\), and since \(A\) is a right contracted ideal, \(r_0\in A\) (by the analogue of Lemma 4 for the ring \(\bar R_{(S)}\)). Hence \(a^ex=[\varphi(\bar s)]^{-1}[\varphi(s_0)]^{-1}\varphi(r_0)\in R_{(S)}\varphi(A)=A^e\). The lemma is proved.

Theorem 2. If the ring \(R\) has rings \(R_{(P)}\) and \(\bar R_{(P)}\), and \(R_{(P)}\cong \bar R_{(P)}\) over \(R\), then the ring \(R_{(P)}\) \((\bar R_{(P)})\) is local.

Proof. If \(sr-p\in I\), where \(s\in S\), \(p\in P\), and \(r\in R\), then \(sr-p\in P\), since \(I\subset P\). Then \(sr\in P\) and, since \(s\notin P\), it follows that \(r\in P\). Thus the ideal \(P\) is right contracted. It is checked analogously that the ideal \(P\) is left contracted. Hence, by Lemma 5, it follows that the left ideal \(P^e=R_{(P)}\varphi(P)\) is two-sided in the ring \(R_{(P)}\), and by the analogue of this lemma for the ring \(\bar R_{(P)}\), the right ideal \(P^e=\varphi(P)\bar R_{(P)}\) is two-sided in the ring \(\bar R_{(P)}\). After this, Theorem 2 is a consequence of Theorem 1 and of its analogue for the ring \(\bar R_{(P)}\).

If \(R\) is a commutative integral domain, then the results of Theorems 1 and 2 coincide with the results set forth in the book of Hodge and Pedoe (\(^{2}\), p. 55).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
2 XI 1959

REFERENCES

\(^{1}\) V. P. Elizarov, Uspekhi Mat. Nauk, 14, no. 5 (89), 207 (1959).
\(^{2}\) V. Hodge, D. Pedoe, Methods of Algebraic Geometry, 3, IL, 1955.