| Summary: | Omega rings (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula>-rings) (and other related structures) are lattice-valued structures (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula> being the codomain lattice) defined on crisp algebras of the same type, with lattice-valued equality replacing the classical one. In this paper, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula>-ideals are introduced, and natural connections with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula>-congruences and homomorphisms are established. As an application, a framework of approximate solutions of systems of linear equations over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula>-fields is developed.
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