# ideal

### adjective

1. conforming to an ultimate standard of perfection or excellence; embodying an ideal

Similar word(s): perfect

2. constituting or existing only in the form of an idea or mental image or conception

*- a poem or essay may be typical of its period in idea or ideal content*

Similar word(s): abstract

3. of or relating to the philosophical doctrine of the reality of ideas

Similar word(s): idealistic

#### Sentences with ideal as an adjective:

*- the ideal theory or philosophy*

*- ideal point*

*- An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.*

### noun

1. the idea of something that is perfect; something that one hopes to attain

Definition categories: thought, idea

2. model of excellence or perfection of a kind; one having no equal

Similar word(s): apotheosis, nonesuch, nonpareil, nonsuch, paragon, saint

Definition categories: person, model

#### Sentences with ideal as a noun:

*- Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny - Carl Schurz*

*- If (1) the empty set were called a "small" set, and (2) any subset of a "small" set were also a "small" set, and (3) the union of any pair of "small" sets were also a "small" set, then the set of all "small" sets would form an ideal.*

*- Let be the ring of integers and let be its ideal of even integers. Then the quotient ring is a Boolean ring.The product of two ideals and is an ideal which is a subset of the intersection of and . This should help to understand why maximal ideals are prime ideals. Likewise, the union of and is a subset of .*