Idea or ideal?

noun An abstract archetype of a given thing, compared to which real-life examples are seen as imperfect approximations; pure essence, as opposed to actual examples

noun An image of an object that is formed in the mind or recalled by the memory

The mere idea of you is enough to excite me

noun More generally, any result of mental activity; a thought, a notion; a way of thinking

noun A conception in the mind of something to be done; a plan for doing something, an intention

I have an idea of how we might escape

noun A purposeful aim or goal; intent

Yeah, that's the idea

noun A vague or fanciful notion; a feeling or hunch; an impression

He had the wild idea that if he leant forward a little, he might be able to touch the mountain-top

adjective Pertaining to ideas, or to a given idea

noun A perfect standard of beauty, intellect etc., or a standard of excellence to aim at

adjective Existing only in the mind; conceptual, imaginary

noun A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring

Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring…

adjective Optimal; being the best possibility

noun A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins)

adjective Perfect, flawless, having no defects

noun A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection

Formally, an ideal I of a given set X is a nonempty subset of the powerset #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI…