Ideal vs. Prime — What's the Difference?

This is an ideal opportunity to save money

The swimming pool is ideal for a quick dip

Our prime consideration is for the children's safety. See Usage Note at perfect.

In an ideal world, we might have made a different decision

Prime real estate.

You're my ideal of how a man should be

A prime candidate for the study.

The ideal of national unity.

The restaurant is considered the ideal in fine dining.

The prime action of the drug.

"those who regarded even a rhetorical commitment to the ideal of disarmament as irresponsibly utopian" (James Carroll).

An ideal work environment.

Athletes in the prime of their lives.

An ideal world where everything works out fine.

This car is definitely past its prime.

An ideal notion of how businesses are run.

Guard dogs primed for attack.

The ideal forms.

Our prime concern here is to keep the community safe.

The ideal theory or philosophy

Both the English and French governments established prime meridians in their capitals.

Ideal point

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

This is a prime location for a bookstore.

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

Thirteen is a prime number.

Let $\backslash mathbb\{Z\}$ be the ring of integers and let $2\backslash mathbb\{Z\}$ be its ideal of even integers. Then the quotient ring $\backslash mathbb\{Z\}\; /\; 2\backslash mathbb\{Z\}$ is a Boolean ring.

The product of two ideals $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is an ideal $\backslash mathfrak\{a\; b\}$ which is a subset of the intersection of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is a subset of $\backslash mathfrak\{a\; +\; b\}$.

Formally, an ideal $I$ of a given set $X$ is a nonempty subset of the powerset $\backslash mathcal\{P\}(X)$ such that: $(1)\backslash \; \backslash emptyset\; \backslash in\; I$, $(2)\backslash \; A\; \backslash in\; I\; \backslash and\; B\; \backslash subseteq\; A\backslash implies\; B\backslash in\; I$ and $(3)\backslash \; A,B\; \backslash in\; I\backslash implies\; A\backslash cup\; B\; \backslash in\; I$.

The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.

There will always be a wide interval between practical and ideal excellence.

The ideal is to be attained by selecting and assembling in one whole the beauties and perfections which are usually seen in different individuals, excluding everything defective or unseemly, so as to form a type or model of the species. Thus, the Apollo Belvedere is the ideal of the beauty and proportion of the human frame.

3 is a prime.

A poem or essay may be typical of its period in idea or ideal content

I'm threatening to build a prime here.

You'll have to press this button twice to prime the fuel pump.

I need to prime these handrails before we can apply the finish coat.

To prime a witness

The boys are primed for mischief.

To prime trees

She was not the prime cause, but I myself.

His starry helm, unbuckled, showed him primeIn manhood where youth ended.

In the very prime of the world.

Hope waits upon the flowery prime.

Give him always of the prime.

Early and late it rung, at evening and at prime.

They sleep till that it was pryme large.

Night's bashful empress, though she often wane,As oft repeats her darkness, primes again.

Prime a cannon

Prime a mine

Prime a car engine

An architect of premier rank

The prime minister

Prime mover

Choice wines

Prime beef

Prize carnations

Quality paper

Select peaches

Prime number

Our manhood's prime vigor