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 -

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 $\backslash mathbb\{Z\}$ 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 $\backslash mathfrak\{a\; +\; b\}$.

Noun

We profess to have embraced a religion which contains the most exact rules for the government of our lives.

A judicious artist will use his eye, but he will trust only to his rule .

There's little can be said in 't; 'Tis against the rule of nature.

Obey them that have the rule over you.

His stern rule the groaning land obeyed.

My rule is to rise at six o'clock.

This uncivil rule ; she shall know of it.

(Wharton)

a rule for extracting the cube root