Hi Elsy! I assume that by real time applications you mean practical applications? If not, please correct me.

One simple application is that we use the quotient ring of the integers by an ideal every time we use a 12- or 24-hour clock, similarly with days of the week. There are applications in number theory, for example in studying Diophantine equations (including their first use in an attempt to prove Fermat's Last Theorem). They also have practical uses in cryptography and coding theory.

In a ring x+ey

x,y real and ee=0 the ye part

Is an ideal.

Then

f(x+ye)=f(x) + ey Df(x)

Is a way to define the derivative Df(x).

Note

ring element* ideal = ideal

A possible application of math Ideals is to use them to represent "societal ideals". Such "ideals" are tenets of Society and their math Ideals can be used to factor the complicated Ring called Society into its more essential "congruence classes", i.e., those sharing similar tenets/beliefs. Such is elaborated in my paper, "Society's 'Ring of Truth' ".

Dear John A. Shuster,

You have done a nice research, I think.

Thank you.

Dear Muhammad, Thanks your comment. Your possible qualification, "I think", may be in reference to my assumption that Society's essence can be captured in an algebraic ring. Yet, the parallels in words: ideal, congruence (in thinking), unity group, "other- groups, non-conforms, etc. seem to make a powerful analogy to societal structures. I took a bit of a "leap"!