ideals occur in several algebraic structures. They play a similar rôle to normal subgroups in groups in that they allow the concept of quotient structures, isomorphism theorems etc.

A right ideal I of a ring R is a nonempty subset of R such that

• a+b in I for a,b in I
• I is closed under right multiplication. i.e. for each a in I and r in R we have ar in I.

A left ideal is defined similarly, except that it has to be closed under left multiplication.

A two-sided ideal (or just ideal) is a right and left ideal of a ring.

For commutative rings there is no difference between the concepts of right,left and two-sided ideals.

Examples of ideals

• If R is any ring then R is itself an ideal of R. So is {0}.
• If a is an element of R then
aR = {ar: r in R}
is a right ideal of R. More generally if a1,...,an are finitely many elements of R then
a1R +...+ anR = {a1r1+...+anrn: each ri in R}
is a right ideal of R.
• Let Z be the ring of integers. Then nZ (i.e. all integer mutiples of n) is an ideal.
• Let x,y be complex numbers then the set of all 2x2 complex matrices of the form
``` --   --
| xa xb |
| yc yd |
--   --
```
with a,b,c,d complex numbers, form a right ideal of the ring of complex 2x2 matrices.

If I and J are right ideals then so is I+J which consists of all sums i+j with i in I and j in J. If I and J are ideals then so is IJ which consists of all finite sums i1j1+...+ikjk, with ir in I and js in J. If S is a subset of R the right ideal it generates is denoted by SR and consists of all finite sums i1j1+...+ikjk, with ir in S and js in R. Note that SR is a right ideal of R. Finally, if Ij is a family of right ideals then the sum of this family consists of all finite sums ij1 +...+ ij1 with ijk in Ijk. The sum is again a right ideal. If a is an element of R then the ideal it generates RaR consists of all finite sums r1as1 + ... + rtast with ri and si in R. This is an ideal. If S is a subset of R then the ideal of R it generates is the sum of the ideals RaR, with a in S. We sometimes write (a1,...,an) for the ideal generated by {a1,...,an}.

During the late 70's and early 80's, if you lived in the Philadelphia Tri-State Area, you must have certainly seen the ubiquitous advertising for Ideal.

Ideal is, or rather, was a women's clothing store that took residence in one of those Quonset Huts that you can see on Gomer Pyle, U.S.M.C. Their clothing was of the inexpensive polyester variety. I'm talking the True Polyester-- the thick, awful against the skin, coarse kind, not the new and nifty-keen silk microfiber kind.

In any case, Ideal would place their ads on UHF stations mid-day and late at night. They were pretty cheesy. The production values were considerably less than ideal. However, they have a super-catchy jingle.

I really can't due justice to the tune due to the nature of Everything, but try to imagine a really upbeat 50's commercial jingle:

If you've got a passion for fashion,
And you've got a craving for savings,
Take the wheel of your automobile,
And swing on down to-- Ideal!

It's little more than a Quonset Hut,
But if you like great savings,
Then, you know what?
Take the Wheel of your automobile,
And swing on down to-- Ideal!

I*de"al (?), a. [L. idealis: cf. F. id'eal.]

1.

Existing in idea or thought; conceptional; intellectual; mental; as, ideal knowledge.

2.

Reaching an imaginary standard of excellence; fit for a model; faultless; as, ideal beauty.

Byron.

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

3.

Existing in fancy or imagination only; visionary; unreal.

"Planning ideal common wealth."

Southey.

4.

Teaching the doctrine of idealism; as, the ideal theory or philosophy.

5. Math.

Imaginary.

Syn. -- Intellectual; mental; visionary; fanciful; imaginary; unreal; impracticable; utopian.

I*de"al (?), n.

A mental conception regarded as a standard of perfection; a model of excellence, beauty, etc.

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. Fleming.

Beau ideal. See Beau ideal.

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