Journal of Mathematical Research with Applications May, 2017, Vol. 37, No. 3, pp. 267{273 DOI:10.3770/j.issn:2095-2651.2017.03.003 Http://jmre.dlut.edu.cn Commutative L∗-Rings Jingjing MA1;∗, Yuehui ZHANG2 1. Department of Mathematics and Statistics, University of Houston-Clear Lake, Houston TX 77058, USA 2. Department of Mathematics, School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P. R. China Abstract We show that for an integral domain or a commutative local ring, it is an L∗-ring if and only if it is an O∗-ring. Some general conditions are also proved for a commutative ring that cannot be L∗. Keywords division closed; lattice order; partial order; total order; F ∗-ring; O∗-ring MR(2010) Subject Classification 06F25 1. Introduction In this paper, all rings are commutative and torsion-free as a group. A ring is called unital if it has an identity element. We review a few definitions and the reader is referred to [1,2] for more information on partially ordered rings and lattice-ordered rings (`-rings). For a partially ordered ring R, the positive cone is defined as R+ = fr 2 R j r ≥ 0g. Let ≤ and ≤0 be two partial orders on a ring R. We say that the partial order ≤0 extends the partial order ≤, if for any a; b 2 R, a ≤ b implies that a ≤0 b. Let P and P 0 be the positive cones of ≤ and ≤0. Then ≤0 extends ≤ just means P ⊆ P 0. A ring R is called an L∗-ring (O∗-ring), if each partial on R can be extended to a lattice order (total order) on R. A ring R is called a dir-L∗-ring (dir-O∗-ring) if each directed partial order on R can be extended to a lattice order (total order) [3]. Clearly an O∗-ring is L∗ and dir-L∗. For a characterization of O∗-rings, the reader is referred to [4]. A partially ordered ring R is called division closed if for any a; b 2 R, ab > 0 and a > 0 implies that b > 0. Each totally ordered ring is clearly division closed. In [3], a ring is defined to be consistently L∗ if every partial order that is division closed can be extended to a lattice order that is division closed, and a ring is consistently O∗ if each partial order that is division closed can be extended to a total order. A partially ordered ring R is called regular division closed if for any a; b 2 R, ab > 0 and a > 0 is a regular element implies that b > 0. If a partially ordered ring is a domain, then clearly division closed and regular division closed are equivalent. Received June 16, 2016; Accepted November 2, 2016 Supported by the National Natural Science Foundation of China (Grant No. 11271275) and the Natural Science Foundation of Shanghai Municipal (Grant No. 13ZR1422500). * Corresponding author E-mail address: [email protected] (Jingjing MA); [email protected] (Yuehui ZHANG) 268 Jingjing MA and Yuehui ZHANG Recently it is proved that a field is O∗ if and only if it is L∗, if and only if it is consistently L∗, and a field is dir-O∗ if and only if it is dir-L∗ (see [5, Theorems 6,7]). In this paper, we continue the study on L∗-rings. 2. Main results Let R be a commutative partially ordered ring with the positive cone that contains regular elements of R. Recall that an element a of R is called a regular element if ab = 0 implies that b = 0 for any b 2 R. Define + + P = fa 2 R j 9 a regular element a1 2 R such that a1a 2 R g: Theorem 2.1 Let R be a commutative partially ordered ring with R+ containing regular element. The P defined above is a partial order on R to make R a partially ordered ring that is regular division closed, and P extends R+. + Proof It is clear that P + P ⊆ P , PP ⊆ P . Suppose that a 2 P \ −P . Then a1a 2 R and + + a2(−a) 2 R for some regular elements a1; a2 2 R , so a1a2a = 0. Thus a = 0 since a1 and a2 are regular. Hence P \ −P = f0g. Therefore P is a partial order on R. + Suppose that ab > 0 and a > 0 with respect to P , and a is regular. Then b1(ab) = c 2 R and + + + a1a = d 2 R for some regular elements a1; b1 2 R , and hence (b1d)b = b1(ab)a1 = ca1 2 R . + Since a1 and a are regular, d is regular, so b1d 2 R is regular. Thus b 2 P . Therefore P is regular division closed. Let z 2 R+ be a regular element. For each a 2 R+, za 2 R+ implies that a 2 P , so R+ ⊆ P . Recall that R is called a domain if for any a; b 2 R, ab = 0 implies that a = 0 or b = 0. The following result is an immediate consequence of Theorem 2.1. Corollary 2.2 Let R be a commutative domain and a partially ordered ring with R+ =6 f0g. Then R+ can be extended to a partial order on R that is division closed. Theorem 2.3 Let R be an integral domain. (1) R is consistently L∗ if and only if R is O∗, and hence R is L∗ if and only if R is O∗. (2) R is dir-L∗ if and only if R is dir-O∗. Proof We first notice that a unital `-ring that is division closed must be an almost f-ring [2, Theorem 4.22]. (1) Suppose that R is consistently L∗ and P is the positive cone of a partial order on R. + + Define P1 = Z 1 + P , where Z is the set of all nonnegative integers. Then P1 is a partial order on R, P ⊆ P1 and 1 2 P1. We leave the verification of those facts to the reader. By Corollary 2.2, P1 is extended to a partial order P2 on R that is division closed. Thus P2 is extended to a lattice order P 0 that is division closed by the assumption. Then (R; P 0) is a totally ordered ring. So R is O∗. Now suppose that R is L∗. We show that R is consistently L∗. Then R is O∗ by previous Commutative L∗-rings 269 paragraph. Let P be the positive cone of a partial order on R that is division closed. By Zorn's Lemma, P is contained in a maximal partial order PM . Certainly 1 2 PM . By Corollary 2.2, ∗ ∗ PM is division closed, and PM is a lattice order since R is L . Thus R is consistently L . (2) The proof is similar to (1) and hence omitted. We look at an example. Example 2.4 Let F be an integral domain and R = F [x1; :::; xn] be the polynomial ring over F with n variables. Then R is not an L∗-ring. Otherwise R is O∗ by Theorem 2.3, so by [6] each element in R is algebraic over Z, where Z is the ring of integers, a contradiction. Actually any integral domain that contains an element which is not algebraic over Z is not L∗ since it is not O∗. Let R be an `-ring. Recall that an element a 2 R+ is called an f-element (d-element) if for any x; y 2 R, x ^ y = 0 implies that ax ^ y = 0 (ax ^ ay = 0). An `-ring is called an f-ring (d-ring) if each positive element is an f-element (d-element). A ring R is called F ∗ if each partial order can be extended to a lattice order to make R into an f-ring. A d-ring must be regular division closed. In fact, if ab > 0 and a > 0 is regular in a d-ring, then ab = jabj = ajbj implies that b = jbj > 0. Theorem 2.5 Let R be a unital commutative local ring and an `-ring. Then R is regular division closed if and only if R is a d-ring. Proof Assume that R is regular division closed and M is the unique maximal ideal of R. We first notice that R+ must contain a unit, otherwise 1 = 1+ − 1− 2 M. As a consequence, 1 > 0. Take a 2 R+. If a 62 M, then a is a unit, so aa−1 = 1 > 0 implies that a−1 > 0 since R is regular division closed. Then a is a d-element by [2, Theorem 1.2]. If a 2 M, 1 + a 62 M, so 1 + a is a d-element. It follows that a is a d-element since 0 ≤ a ≤ 1 + a. Therefore each positive element is a d-element and R is a d-ring. Corollary 2.6 Let R be a unital commutative local ring. (1) R is L∗ if and only if it is O∗. (2) R is consistently L∗ if and only if R is O∗. (3) R is dir-L∗ if and only if R is dir-O∗. Proof (1) Suppose that R is a unital commutative local ring that is L∗. Let P be a maximal partial order on R. Then 1 2 P and P is a lattice order. By Theorem 2.1, P is regular division closed, so by Theorem 2.5, (R; P ) is a d-ring. It is well-known that a unital d-ring must be an f-ring, and hence (R; P ) is an f-ring.