Question

Let \(\rho_{i}: R_{i} \rightarrow R_{i}^{\prime},\) for \(i=1, \ldots, k,\) be ring homomorphisms. Show that the map $$ \begin{aligned} \rho: \quad R_{1} \times \cdots \times R_{k} & \rightarrow R_{1}^{\prime} \times \cdots \times R_{k}^{\prime} \\ \left(a_{1}, \ldots, a_{k}\right) & \mapsto\left(\rho_{1}\left(a_{1}\right), \ldots, \rho_{k}\left(a_{k}\right)\right) \end{aligned} $$ is a ring homomorphism.

Short answer

Question: Prove that the map \(\rho: R_1\times \cdots \times R_k \rightarrow R_1' \times \cdots \times R_k'\) which maps \((a_1,\ldots, a_k) \mapsto (\rho_1(a_1),\ldots, \rho_k(a_k))\) is a ring homomorphism. Answer: To show that \(\rho\) is a ring homomorphism, we need to prove three properties: the addition property (\(\rho(a+b) = \rho(a) + \rho(b)\)), the multiplication property (\(\rho(ab) = \rho(a)\rho(b)\)), and the identity mapping (\(\rho(1) = 1'\)). By verifying each property using the given individual ring homomorphisms \(\rho_i: R_i \rightarrow R_i'\) for \(i=1,\ldots, k\), we can conclude that \(\rho\) is a ring homomorphism.

Step-by-step solution

Proving addition property of the ring homomorphism

Let \(a = (a_1,\ldots, a_k)\) and \(b = (b_1,\ldots, b_k)\) be elements in the ring \(R_1\times \cdots \times R_k\). Then, their sum is given by \(a + b = (a_1 + b_1, \ldots, a_k + b_k)\). Now, we can evaluate the left-hand side of the addition property: $$\rho(a + b) = \rho(a_1+b_1,\ldots,a_k+b_k) = (\rho_1(a_1 + b_1),\ldots,\rho_k(a_k + b_k)).$$ As all \(\rho_i\) are ring homomorphisms, we have \(\rho_i(a_i + b_i) = \rho_i(a_i) + \rho_i(b_i)\) for all \(i\). Thus, we get: $$\rho(a + b) = (\rho_1(a_1)+\rho_1(b_1), \ldots, \rho_k(a_k)+\rho_k(b_k)) = \rho(a) + \rho(b).$$

Proving multiplication property of the ring homomorphism

Now, let's consider the product \(ab = (a_1b_1,\ldots,a_kb_k)\). We can compute the left-hand of the multiplication property: $$\rho(ab) = \rho(a_1b_1,\ldots,a_kb_k) = (\rho_1(a_1b_1),\ldots,\rho_k(a_kb_k)).$$ Since all \(\rho_i\) are ring homomorphisms, we have \(\rho_i(a_ib_i) = \rho_i(a_i)\rho_i(b_i)\) for all \(i\). Hence, we get: $$\rho(ab) = (\rho_1(a_1)\rho_1(b_1), \ldots, \rho_k(a_k)\rho_k(b_k)) = \rho(a)\rho(b).$$

Proving the identity mapping of the ring homomorphism

Finally, let's verify that \(\rho\) maps the identity element of \(R_1 \times \cdots \times R_k\) to the identity element of \(R_1' \times \cdots \times R_k'\). The identity element of \(R_1 \times \cdots \times R_k\) is given by the tuple \((1_{R_1},\ldots,1_{R_k})\). Let's compute \(\rho(1)\): $$\rho(1_{R_1},\ldots, 1_{R_k}) = (\rho_1(1_{R_1}),\ldots,\rho_k(1_{R_k})).$$ Since all \(\rho_i\) are ring homomorphisms, we know that \(\rho_i(1_{R_i}) = 1_{R_i'}\) for all \(i\). Thus, we obtain: $$\rho(1_{R_1},\ldots, 1_{R_k}) = (1_{R_1'},\ldots,1_{R_k'}) = 1',$$ where \(1'\) is the identity element in \(R_1' \times \cdots \times R_k'\). With all three properties verified, we can conclude that \(\rho\) is indeed a ring homomorphism.

Key concepts

Ring Theory

Ring theory is a fundamental branch of abstract algebra involving the study of rings—a type of algebraic structure consisting of a set equipped with two binary operations that we usually call addition and multiplication. These operations on the elements of the ring must satisfy certain properties such as associativity, distributivity, and having an additive identity (also known as the zero element) and multiplicative identity (the number one).

One of the key aspects of ring theory is the study of ring homomorphisms. These are functions between two rings that preserve the ring operations, meaning they respect both the addition and multiplication structure of the rings involved. If you imagine two rings as two different languages, a ring homomorphism would be like a translator who accurately translates sentences (elements and their combinations) from one language to another without changing the meaning (structure and properties of operations).

Abstract Algebra

Abstract algebra is the area of mathematics that studies algebraic structures such as groups, rings, and fields. These structures form the language and backdrop for much of modern mathematics and are motivated by solving polynomial equations and understanding symmetry. One of the core activities in abstract algebra is to understand how these structures relate to each other through functions like homomorphisms.

Within this field, ring homomorphisms allow us to map elements from one ring to another while preserving the ring's structure. This concept is crucial for understanding the fundamental connections and similarities between different algebraic systems and for categorizing them into broader families based on shared properties. The exercise provided focuses on ring homomorphisms, a prime example of the kinds of relationships and transformations studied in abstract algebra.

Ring Homomorphisms Properties

Ring homomorphisms have specific properties that render them special functions between rings. Here are the primary properties:

• Preservation of Addition: If \(f: R \rightarrow S\) is a ring homomorphism, then for any elements \(a, b \in R\), it must fulfill that \(f(a+b) = f(a) + f(b)\).
• Preservation of Multiplication: Similarly, for any elements \(a, b \in R\), the homomorphism \(f\) must satisfy \(f(ab) = f(a)f(b)\).
• Identity Mapping: A ring homomorphism will map the multiplicative identity of the domain ring to the multiplicative identity of the codomain ring, so if \(R\) has identity \(1_R\) and \(S\) has identity \(1_S\), then \(f(1_R) = 1_S\).

These properties ensure that the homomorphism aligns with the structure of the rings in question, as demonstrated in the exercise solution where the product ring homomorphism \(\rho\) is shown to adhere to all three key properties.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of mathematical statements. They are rigorous means of demonstrating that a conclusion follows unavoidably from initial assumptions known as axioms. In the realm of ring theory, proving that a certain function is a ring homomorphism involves checking it satisfies the aforementioned properties.

The step-by-step solution provided in the exercise is an example of a mathematical proof. It verifies the properties methodically: first proving that addition in the domain ring leads to addition in the codomain ring when mediated by the homomorphism, then proving the same for multiplication, and finally verifying that identities map to identities. These thorough checks underpin the validity of the conclusion that the function in question is a true ring homomorphism, encapsulating the very essence of mathematical proof.