Shirshova, E. E.

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ON PARTIALLY ORDERED RINGS
E.E. Shirshova
Moscow, Russia
Let G be a partially ordered additive group [1]. An element x 2 G is said to be very much
less than y 2 G (x y) if nx 6 y for all n 2 Z. We denote by Ma the set of all g 2 G such that
g  a for some a 2 G, where a > 0.
Definition 1. A ring R is called a partially right K-ordered ring, if it satisfies the conditions:
1) < R;+ > is a partially ordered group; 2) if a > 0 in R, then ar  a for all r 2 R. If the order
of the group < R;+ > is linear, we say that the ring R is a linear right K-ordered ring.
Theorem 2. Let R be a ring, I being an ideal of R satisfying the conditions: 1) the rings
< I;61> and < R=I;62> are partially right K-ordered rings; 2) the conditions a 2 I and a >1 0
imply the inequality ar 61 a for each r 2 R. Then the ring R can be furnished with a partial right
K-order 6, from which the order 61 inherits in the ring I, and the order 62 inherits in the ring
R=I; and I being a convex ideal of R under the order 6.
Let U be a system of ideals of a ring R, including f0g and R. The system of ideals U is called
complete, if for any subsystem of U the following is true: both union and intersection of ideals
forming this subsystem belong to U. An inclusion of ideals A  B are said to be a step, if the
relation A  J  B implies A = J or J = B for each ideal J of R.
Theorem 3 If R is a linear right K-ordered ring, then there exists a complete system of right
ideals in R, where any step of right ideals I  J in this system satisfies the conditions: 1) JR  I;
2) I is a convex ideal of J ; 3) I =Mb for all b 2 J n I, where b > 0.
Theorem 4 Let R be a ring without an identity element, and there exists a complete system
of right ideals in R which satisfies the condition: if I  J is any step of right ideals in this system,
then I is an ideal of J , and the quotient group < J=I;+ > is a torsion-free group. Then R can be
made into a linear right K-ordered ring.
References
1. Fuchs L. Partially ordered algebraic systems. Oxford - London - New York - Paris: Pergamon Press,
1963.


On partially ordered rings

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On partially ordered rings

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