On the lattice of biorder ideals of regular rings

Abstract

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.

Biordered sets of lattices and homogeneous basis

“…(cf. [5]) If P is a partially ordered set and Φ : P → P is an isotone(order preserving) mapping, the Φ will be called normal if (1) imΦ is a principal ideal of P and (2) whenever xΦ = y, then there exists some z ≤ x such that Φ maps the principal ideal P (z) isomorphically onto the principal ideal P (y).…”

“…, N } are independent. Now recall from [1], that given a regular ring R whose biordered set of idempotents E R then the principal biorder ideals ω l (e) [ω r (e)], e ∈ E R is the complemented modular lattice Ω L [Ω R ]. Also we have the following theorem Theorem 2.…”

“…In [2] we extend the biordered set approch from regular semigroups to regular rings by explicitly desceribing the structure of the multiplicative idempotents E R of a regular ring R with two quasi orders ω l and ω r as a bounded and complemented biorderd set . In [1], we describe the biorder ideals generated by elements in E R as {ω l (e) : e ∈ E R } and {ω r (e) : e ∈ E R } and are the complemented modular lattice Ω L and Ω R respectively. Further we also discuss certain interesting propertes of the complemented modular lattice such as perspectivity and independence of the lattce and obtained the necessary and suffient condition at which the complementd modular lattice is of order n.…”

“…It is also seen that on some conditions in E P (L) , the complemented modular lattice L admits a homogeneous basis and this lattice L corresponds to the lattice Ω L the lattice of all ω l -ideals of a ring R (cf. [1]).…”

“…1) if and only if (n j ; v j ) = (v i ; n i ).Proof. For each e ∈ E R , (ω l (1 − e); ω l (e)) is a complementary pair in the lattice Ω L and the mapping ǫ : e → (ω l (1 − e); ω l (e)) for all e ∈ E R is a map of E R into E P (Ω l ) .…”