Question

Let \(\rho: R \rightarrow R^{\prime}\) be a ring homomorphism. Let \(S\) be a subring of \(R,\) and let \(\tau: S \rightarrow R^{\prime}\) be the restriction of \(\rho\) to \(S\). Show that \(\tau\) is a ring homomorphism and that \(\operatorname{Ker} \tau=\operatorname{Ker} \rho \cap S\)

Short answer

Question: Show that if a ring homomorphism ρ: R → R′ exists and a subring S of R exists, then the restriction of ρ to S, τ: S → R′, is also a ring homomorphism. Also, show that the kernel of τ is the intersection of the kernel of ρ and S. Answer: We showed that the restriction of ρ to S, τ: S → R′ satisfies the properties of a ring homomorphism by preserving addition and multiplication. Finally, we proved that Ker(τ) = Ker(ρ) ∩ S, by showing that any element of S in the kernel of ρ is also in the kernel of τ, and vice versa.

Step-by-step solution

Restriction of a function

Let \(\tau: S \rightarrow R^{\prime}\) be the restriction of \(\rho: R \rightarrow R^{\prime}\) to \(S\). In other words, for every \(s \in S\), we have \(\tau(s) = \rho(s)\).

Proving that \(\tau\) is a ring homomorphism

To show that \(\tau\) is a ring homomorphism, we need to prove that it preserves addition and multiplication. Given any \(s_1, s_2 \in S\), we need to check that \(\tau(s_1 + s_2) = \tau(s_1) + \tau(s_2)\) and \(\tau(s_1s_2) = \tau(s_1)\tau(s_2)\). For addition, we have: $$ \tau(s_1 + s_2) = \rho(s_1 + s_2) = \rho(s_1) + \rho(s_2) = \tau(s_1) + \tau(s_2) $$ For multiplication, we have: $$ \tau(s_1s_2) = \rho(s_1s_2) = \rho(s_1)\rho(s_2) = \tau(s_1)\tau(s_2) $$ Since \(\tau\) preserves both addition and multiplication, we conclude that it is a ring homomorphism.

Proving the relationship between the kernels

We need to show that \(\operatorname{Ker}\tau = \operatorname{Ker}\rho \cap S\). This means that any element of \(S\) that is in the kernel of \(\rho\) should also be in the kernel of \(\tau\), and vice versa. Let \(s \in \operatorname{Ker}\tau\), meaning that \(\tau(s) = 0\) in \(R^{\prime}\). Since \(\tau(s)=\rho(s)\), we have \(\rho(s) = 0\). Therefore, \(s \in \operatorname{Ker}\rho\). Since \(s \in S\) by assumption, we have \(s \in \operatorname{Ker}\rho \cap S\). Thus, \(\operatorname{Ker}\tau \subseteq \operatorname{Ker}\rho \cap S\). On the other hand, let \(s \in \operatorname{Ker}\rho \cap S\). This means that \(s \in \operatorname{Ker}\rho\) and \(s \in S\). Since \(s \in \operatorname{Ker}\rho\), we have \(\rho(s) = 0\). Considering the definition of \(\tau\), we get \(\tau(s) = \rho(s) = 0\), so \(s \in \operatorname{Ker}\tau\). Thus, \(\operatorname{Ker}\rho \cap S \subseteq \operatorname{Ker}\tau\). Since \(\operatorname{Ker}\tau \subseteq \operatorname{Ker}\rho \cap S\) and \(\operatorname{Ker}\rho \cap S \subseteq \operatorname{Ker}\tau\), we conclude that \(\operatorname{Ker}\tau = \operatorname{Ker}\rho \cap S\).

Key concepts

Subring

A subring is a smaller ring within a larger ring that retains the properties necessary to be considered a ring itself. Let's dive into this term "subring." For any algebraic structure, such as a ring, a substructure inherits the properties of the original structure. In the context of rings, a subring must itself be a ring, which means it must include the multiplicative identity element (if the larger ring has one), be closed under the operations of addition and multiplication, and for each of its elements, the additive inverse must also belong to the subring.
Moreover, a subring should satisfy:

• Closure under addition: If you take any two elements from the subring and add them, the sum should still be in the subring.
• Closure under multiplication: Multiply any two elements, and the product should be in the subring too.
• Contains zero element: The element 0 (zero) must be present in the subring because the additive identity is crucial for the ring's properties.
• Negation: For each element in the subring, there must also be an additive inverse in the subring.

These properties ensure we maintain a consistent structure similar to the larger ring but on a smaller scale.

Kernel of a Homomorphism

In the world of ring theory, understanding the kernel of a homomorphism is key to grasping how closely connected different algebraic structures are. A ring homomorphism, like \(\rho: R \rightarrow R'\), is a function that respects the operations of the rings. But what's the role of its kernel?
The kernel of the homomorphism \(\rho\) is essentially the set of elements in \(R\) that get mapped to the zero element in \(R'\) by \(\rho\). Mathematically, it can be represented as:
\[ \text{Ker}(\rho) = \{ r \in R \mid \rho(r) = 0 \} \]
Think of the kernel as a measure of the "bending" done by the homomorphism — it contains all elements that \(\rho\) shrinks down to 0.

• The kernel is always a subring of \(R\).
• It is used to determine if the homomorphism is injective (a map with this property is one-to-one): if the kernel only contains the zero element, the homomorphism is injective.
• Furthermore, the kernel gives an insight into the structure of \(R\) after it has been mapped by the homomorphism.

Indeed, the kernel plays a pivotal role in the overarching structure and function of homomorphisms.

Restriction of a Function

Sometimes, we don't need all the context provided by a large set — we just want a slice, a "restriction" if you will, tailored to a subset. This is where the concept of the restriction of a function comes into play. Restricting a function essentially means narrowing its domain. Instead of applying the function to an entire set, we apply it to a specific part — a subring, for instance.
The key idea is to maintain the same functionalities as before, but over a smaller, specified section of the larger domain. In the exercise with the ring homomorphism \(\rho: R \rightarrow R'\), the function is restricted to a subring \(S\), creating a new function \(\tau: S \rightarrow R'\).

• Formally, for any \(s \in S\), \(\tau(s) = \rho(s)\).
• Despite being restricted, \(\tau\) adheres to the homomorphism properties, just as \(\rho\) does.
• This preservation is due to the restriction ensuring that only elements already pertaining to \(S\) are considered, making all operations still valid and applicable.

Restrictions are useful tools for focusing analysis on specific parts of a domain, allowing us to examine behaviors in a controlled environment.