Let k be a field, let R=k[x, y], and let I be the ideal (x, y). (i) Prove that x ⊗y-y ⊗x ∈I ⊗R I

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Let k be a field, let R=k[x, y], and let I be the ideal (x,...

Key Concepts

Tensor Product of Modules

The tensor product is an operation that combines two modules over a ring to form another module, capturing bilinear relationships between them. This construction is essential in various areas including homological algebra, representation theory, and algebraic geometry, particularly when studying extensions and the behavior of modules under base change.

Ideal

An ideal is a special subset of a ring that is closed under addition and absorbs multiplication by elements of the ring. Ideals are used to construct quotient rings and to study the structure of rings, with particular relevance in defining algebraic sets and examining local properties.

Torsion and Torsion-Free Modules

A module is said to be torsion-free if none of its nonzero elements are annihilated by a nonzero element of the ring. This concept is critical in module theory and algebra, as it distinguishes modules with ‘nice’ properties that resemble vector spaces from those having elements that can be ‘killed’ by ring elements, which plays an important role in understanding module structure and singularities.

Field

A field is an algebraic structure in which every nonzero element has a multiplicative inverse. Fields provide the scalars over which vector spaces, modules, and many ring constructions are built, serving as a fundamental basis for much of algebra.

Polynomial Ring

A polynomial ring is a ring formed from all polynomials with coefficients in a given field. Such rings are central to algebraic geometry and commutative algebra, as they serve as coordinate rings of affine spaces and provide examples for studying ideals, factorization, and modules.

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