# Question on Ideals (Algebraic Geometry)

• arunma
In summary, the conversation discusses the concept of one ideal being "strictly bigger" than another in algebra. The speaker clarifies that an ideal I is strictly bigger than an ideal J if J is a proper subset of I, but also asks for the meaning of this in a larger context. They express interest in understanding the geometric and algebraic implications of this concept.
arunma
I have a question for anyone who is knowledgeable in algebra. What does it mean for one ideal to be "strictly bigger" than another?

An ideal I strictly bigger than the ideal J iff J is a proper subset of I.

at least that's what i think.

this is such a trivial question that it must not be the real question. I.e. what is the context for this question and why is it puzzling you?

Yes, I suppose that was a trivial question. Indeed the answer you provided is also given in my book. I suppose what I should have asked is: what does it "mean" for one ideal to be strictly larger than another. In particular, if the terms in one ideal are not divisible by the terms in a second ideal, does this make the first ideal larger?

I'm sorry that I am phrasing my questions rather imprecisely. But I don't take algebra courses very often (I generally favor analysis), so I'm still trying to figure out precisely what ideals are and why they are of interest to mathematicians. I'd be very interested to hear any insight you've got on this.

these quesyions are more interesting:

1) what does it mwean geometrically for an ideL TO BE larger?

2) how does one check LGEBRAICALLY THt AN IDEAL generated by a given finites et of elements, is indeed lakrger thAN AN IDEAL generate dby anoither finite set of elements?

these are not so easy. i know somethings about them but i will answer later.

## 1. What are ideals in algebraic geometry?

Ideals in algebraic geometry are subsets of a polynomial ring that satisfy certain properties. They are used to describe the geometric properties of algebraic varieties.

## 2. How are ideals related to algebraic varieties?

Ideals are intimately related to algebraic varieties in that they define the zero sets of polynomial equations in the corresponding affine or projective space. They also play a crucial role in studying the geometric and algebraic properties of these varieties.

## 3. What are the key properties of ideals?

Ideals have several key properties, including closure under addition and multiplication, as well as the existence of a unique minimal generating set. They also have a well-defined set of associated primes and a Grobner basis, which can be used for solving systems of polynomial equations.

## 4. How are ideals used in practical applications?

Ideals have numerous practical applications, including in coding theory, cryptography, and computer-aided design. They are also used in solving systems of polynomial equations, which arise in various fields such as physics, engineering, and economics.

## 5. What is the relationship between ideals and algebraic varieties in higher dimensions?

The relationship between ideals and algebraic varieties becomes more complex in higher dimensions. In addition to defining the zero sets of polynomial equations, ideals are also used to describe the singularities and other geometric properties of these varieties. They are also crucial in the study of the cohomology and intersection theory of higher-dimensional varieties.

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