Question

Let \(R\) be a subring of \(S, S\) a subring of \(T\). (a) If \(S=\Sigma R v_{i}, T=\Sigma S w_{j}\), show that \(T=\sum R v_{i} w_{j}\). (b) If \(S=R\left[v_{1}, \ldots, v_{n}\right], T=S\left[w_{1}, \ldots, w_{m}\right]\), show that \(T=R\left[v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right]\). (c) If \(R, S, T\) are fields, and \(S=R\left(v_{1}, \ldots, v_{n}\right), T=S\left(w_{1}, \ldots, w_{m}\right)\), show that \(T=\) \(R\left(v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right)\) So each of the three finiteness conditions is a transitive relation.

Short answer

In summary, we have shown that the relation between rings is transitive for the given cases. For case (a), we demonstrated that if \(S = \Sigma R v_i\) and \(T = \Sigma S w_j\), then \(T = \sum Rv_iw_j\). In case (b), it was shown that if \(S = R[v_1, \ldots, v_n]\) and \(T = S[w_1, \ldots, w_m]\), then \(T = R[v_1, \ldots, v_n,w_1, \ldots, w_m]\). Lastly, in case (c), we proved that if \(R, S\) and \(T\) are fields and \(S = R(v_1, \ldots, v_n)\), \(T = S(w_1, \ldots, w_m)\), then \(T = R(v_1, \ldots, v_n, w_1, \ldots, w_m)\).

Step-by-step solution

Understand the notation

Here, \(\Sigma R v_{i}\) means the sum of all elements of form \(r_iv_i\) where \(r_i\) is an element from ring \(R\) and \(v_i\) is from the set of elements forming ring \(S\). Similarly, \(\Sigma S w_{j}\) means the sum of all elements of form \(s_jw_j\) where \(s_j\) is an element from ring \(S\) and \(w_j\) is from the set of elements forming ring \(T\).

Prove the relation between rings

In order to prove that \(T= \sum R v_{i} w_{j}\), we have to show that any element \(t \in T\) can be written as a sum of elements of form \(rv_iw_j\) where \(r \in R\), \(v_i\) is from the set of elements forming ring \(S\), and \(w_j\) is from the set of elements forming ring \(T\). Since \(T=\Sigma S w_{j}\), we can represent any element \(t \in T\) as \(t=\sum s_j w_j\). Now, we know that \(S = \Sigma R v_{i}\). Therefore, we can represent any element \(s_j \in S\) as \(s_j = \sum r_i v_i\).

Combine the representations of t and s_j

Now, substitute the expression of \(s_j\) in the expression for \(t\): \(t=\sum (\sum r_i v_i) w_j = \sum r_i v_i w_j\). Therefore, we have shown that any element \(t\in T\) can be written as a sum of elements of form \(r v_i w_j\). This implies that \(T=\sum R v_{i} w_{j}\). ------------------------------------------------------------------------------------------------------------------------ (b) If \(S=R\left[v_{1}, \ldots, v_{n}\right], T=S\left[w_{1}, \ldots, w_{m}\right]\), show that \(T=R\left[v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right]\).

Understand the notation

The notation \(R\left[v_1, \ldots, v_n\right]\) means the smallest subring of \(S\) containing all elements \(v_1, \ldots, v_n\) and all elements of ring \(R\). Similarly, \(S\left[w_1, \ldots, w_m\right]\) means the smallest subring of \(T\) containing all elements \(w_1, \ldots, w_m\) and all elements of ring \(S\).

Show T is the smallest subring containing elements from R, v's and w's

In order to show that \(T=R\left[v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right]\), we need to show that \(T\) is the smallest subring containing all elements of the sets \(R\), \(\{v_1, \ldots, v_n\}\) and \(\{w_1, \ldots, w_m\}\). Any element in \(T\) can be represented as a sum of the form \(\sum s_i w_j = \sum (\sum r_k v_l) w_j = \sum r_k v_l w_j\), where \(s_i \in S\) are of the form \(\sum r_k v_l\), and \(r_k \in R\), \(v_l\) is from the set of elements {\(v_1, \ldots, v_n\)}, and \(w_j\) is from the set of elements {\(w_1, \ldots, w_m\)}. Since \(T\) is by definition the smallest subring containing all elements of \(S\) and \(\{w_1, \ldots, w_m\}\), and we have shown that elements in \(T\) can also be expressed as sums of elements from the sets \(R\), \(\{v_1, \ldots, v_n\}\) and \(\{w_1, \ldots, w_m\}\), we conclude that \(T=R\left[v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right]\). ------------------------------------------------------------------------------------------------------------------------ (c) If \(R, S, T\) are fields, and \(S=R\left(v_{1}, \ldots, v_{n}\right), T=S\left(w_{1}, \ldots, w_{m}\right)\), show that \(T=\) \(R\left(v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right)\).

Recognize the fields property

The given information is that \(R, S, T\) are fields, which means they are commutative rings such that every nonzero element has a multiplicative inverse.

Combine step 1 and (b)'s result

Now, from part (b) of this exercise, we know that \(T=R\left[v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right]\). Since \(T\) is a field and contains all elements of \(R\), \(\{v_1, \ldots, v_n\}\) and \(\{w_1, \ldots, w_m\}\), it follows that: \(T = R\left(v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right)\) Hence, the finiteness condition is transitive for fields as well.

Key concepts

Subring Theory

Subring theory is a fundamental concept in algebraic geometry and commutative algebra. It deals with the idea of a subring, which is a subset of a ring that is itself a ring under the same addition and multiplication operations as the larger ring. This becomes essential when trying to organize complex algebraic structures.

In general, for any ring, a subring must include the multiplicative identity of the larger ring, and it must be closed under subtraction and multiplication.

• Closure: For any elements a and b in a subring, both a-b and ab must also be in the subring.
• Identity: The subring must contain the ring's identity element, which is typically 1 in the case of commutative rings.

The exercise makes use of subring theory when it talks about expressing a ring T as a sum of elements formed by ring R and elements v and w. This is an example of how subrings can build upon one another to form larger and more complex rings.

Field Theory

Field theory extends the concept of arithmetic beyond the realm of integers and real numbers. In fields, every non-zero element has a multiplicative inverse, making fields a key concept within commutative algebra.

Fields are one of the most structured algebraic systems. They are commutative rings where division (except by zero) is possible. This means for any non-zero element 'a' in a field, there exists an element 'b' such that ab = 1.

The problem’s use of fields illustrates how rings can be specialized. When discussing fields R, S, and T, and noting that these fields can be combined to form extended fields like R(w,v), it's important to remember:

• Closure under addition and multiplication, just like rings
• Existence of additive and multiplicative inverses

Fields appear often in algebraic geometry since they provide the structure necessary for solving equations and understanding geometric objects algebraically.

Ring Extensions

Ring extensions occur when a smaller ring is extended by adding new elements or variables, thus forming a larger ring. This concept is crucial in understanding the construction of new algebraic systems.

In the provided exercise, the concept of ring extensions is used extensively when moving from ring R to ring S by adding elements v_i, and from S to T by adding elements w_j. This results in T being a ring extension of R, incorporating both sets of elements. Key points include:

• Adjoining new elements: Such as adding a variable to form polynomial rings.
• Preserving ring operations: The new elements must conform to the existing ring structure's operations.

Ring extensions are crucial for simplifying complex structures by working in stages, like breaking down any polynomial into linear factors over some suitably extended ring.

Commutative Algebra

Commutative algebra studies commutative rings, which are foundational structures in algebraic geometry. A commutative ring is one where the multiplication operation is commutative; that is, the order in which two elements are multiplied doesn't matter (e.g., ab = ba).

This branch of mathematics also looks at the ideals and modules over these rings, which help in understanding how different algebraic structures relate to one another. In the exercise, commutative algebra helps to understand how rings R, S, and T interrelate. It gives shape to:

• Local properties of rings: Enabling the categorization and mapping of geometric data.
• Forming algebraic objects: Such as varieties and schemes.

Through commutative algebra, complex algebraic structures can be better understood, offering a robust framework necessary for advancing in algebraic geometry.