For every commutative ring k, prove that the direct product R ×S is the categorical product in C

step 1 In this problem, late, A is equal to A11, A12, A13, A1N, 0, A2, A23, A1N, 0, A2, A23, A3, A3N, 0

For every commutative ring k, prove that the direct product...

For every commutative ring $k$, prove that the direct product $R \times S$ is the categorical product in ComAlg $_{k}$ (in particular, direct product is the categorical product in ComAlg $_{Z}=$ ComRings).

Key Concepts

Category of Commutative Algebras

The category of commutative algebras over a given base ring encompasses objects that are commutative rings endowed with a compatible module structure over the base ring, and whose morphisms respect both the ring and module structures. Understanding this category is crucial since it frames the context in which the direct product is shown to be the categorical product by examining the algebra homomorphisms and structure-preserving maps between objects.

Direct Product of Algebraic Structures

The direct product in algebra is a construction that combines two algebraic structures (such as rings or algebras) into a new structure whose underlying set is the Cartesian product of the two sets, with operations performed coordinatewise. In the setting of commutative algebras over a ring, this construction not only carries an algebraic structure but also inherently satisfies the required universal property for being the categorical product.

Categorical Product

In category theory, the categorical product of two objects is another object equipped with projection maps to each factor such that any object with maps to the factors uniquely factors through it. This abstract definition captures the essence of ‘product’ in various mathematical categories by emphasizing the universal property, ensuring that the product is uniquely determined up to isomorphism.

Universal Mapping Property

The universal mapping property is a key formulation in category theory which states that for any object with morphisms to the factors of a product, there exists a unique morphism to the product object making the entire diagram commute. This property guarantees that the product object serves as the most efficient or 'universal' way to capture the morphisms into the factors, which is essential in proving that a given construction is indeed the categorical product.

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