# Algebras generated as commutative algebras.

• Kreizhn
In summary, the conversation discusses the concept of a commutative R-algebra being finitely generated as an algebra over R. The difference between being generated as an R-algebra and a commutative R-algebra is also addressed. The key point is that for a commutative R-algebra to be finitely generated, there must be a surjection from A[T_1,...,T_n] to S, whereas for it to be finitely generated as an algebra over R, there must be a surjection from A<T_1,...,T_n> to S. The conversation concludes with the task of proving the equivalence of these two notions for a commutative R-algebra.
Kreizhn

## Homework Statement

Let R be a commutative ring. Prove that a commutative R-algebra S is finitely generated as an algebra over R if and only if it is finitely generated as a commutative algebra over R.

## The Attempt at a Solution

I'm really not even sure what this is asking. I know that for an R-algebra S, we can view it as being generated as an R-module and as an R-algebra, and that they do not generally agree. However, what's the difference between being generated as an R-algebra and being generated as a commutative R-algebra?

Hi Kreizhn!

I'm not sure, but I'll have a go at it. I think that S is finitely generated as a commutative A-algebra if there is a surjection

$$A[T_1,...,T_n]\rightarrow S$$

However, S is finitely generated as an A-algebra if there is a surjection

$$A<T_1,...,T_n>\rightarrow S$$

Where $A<T_1,...,T_n>$ are polynomials in n variables, but the variables don't commute. So we don't have $T_1T_2=T_2T_1$.

What you need to show is that if S is commutative, then these two notions are equivalent.

Okay thanks. I'll give it a shot and see what I can come up with.

## 1. What is meant by "algebras generated as commutative algebras"?

"Algebras generated as commutative algebras" refers to a type of algebraic structure where the elements of the algebra are generated by a set of elements that commute with each other. This means that the order in which the elements are multiplied does not affect the outcome, making the algebra commutative.

## 2. How are algebras generated as commutative algebras different from other types of algebras?

Algebras generated as commutative algebras are different from other types of algebras because they have a specific set of elements that commute with each other. In other types of algebras, such as non-commutative algebras, the order of multiplication does affect the outcome.

## 3. What are some examples of algebras generated as commutative algebras?

Some examples of algebras generated as commutative algebras include polynomial algebras, power series algebras, and group algebras. These algebras all have elements that commute with each other, making them commutative algebras.

## 4. How are algebras generated as commutative algebras used in mathematics?

Algebras generated as commutative algebras are used in various areas of mathematics, including abstract algebra, algebraic geometry, and number theory. They are also used in applications such as coding theory, cryptography, and physics.

## 5. What are the implications of studying algebras generated as commutative algebras?

Studying algebras generated as commutative algebras allows for a better understanding of the properties and structures of commutative algebras, as well as their applications in different fields. It also helps to develop new techniques and methods for solving problems in mathematics and other disciplines.

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