Prove that HomR(P, R) ≠{0} if P is a nonzero projective left R module.

Name: Problem 11, Chapter 3: An Introduction to Homological Algebra
Uploaded: 2020-09-21T13:36:44-08:00
Duration: 6 min 40 s
Channel: John Gehad
Description: Prove that HomR(P, R) ≠{0} if P is a nonzero projective left R module.

Prove that HomR(P, R) ≠{0} if P is a nonzero projective left...

Key Concepts

Projective Module

A projective module is one that has the property that every epimorphism onto it splits, or equivalently, it can be expressed as a direct summand of a free module. This property is significant in module theory because it guarantees that the module behaves well with respect to exact sequences and Hom functors, making it easier to analyze and apply in various algebraic contexts.

Hom Functor

The Hom functor, in this case Hom_R(P, R), associates to a module P the set of all R-linear maps from P to R. It is an important construction because it transforms modules into dual objects that can reveal deeper structural properties. Nontrivial elements in this set indicate interesting duality phenomena in the category of modules over a ring.

Direct Summand

A direct summand of a module is a submodule such that the original module can be written as a direct sum of this submodule and another complementary submodule. The property that every projective module is a direct summand of some free module is key in many proofs, including proving the existence of nonzero homomorphisms from the module to the ring.

Nonzero Module

The concept of a nonzero module is fundamental because many module properties are nontrivial only when the module is not the zero module. In this context, the nonzero nature of the projective module ensures that when it is considered as a direct summand of a free module, it contributes nontrivially, thereby guaranteeing that the set of homomorphisms to the ring is nonempty.

Transcript

00:01 Problem number 75.

00:05 The curl of 1 over the norm of r multiply p equals negative pr over the norm of r power p plus 2.

00:16 The dot product of the curl and negative pr over norm of r power p plus 2 equal with sorry the curl equal partial derivative with respect to x plus p partial derivative with respect to y plus partial derivative with respect to z okay therefore we can now substitute in this equation dot negative p r sorry x over x square plus y square plus z square are p plus 2 over 2 i plus negative p y over x square plus y square plus z square power p plus two over two j plus negative p multiply z over x squared plus y square plus z square power p plus two over two this is equal to partial derivative with respect to x of negative p x over norm of r power p plus 2 plus partial derivative with respect to y of negative b y over norm of r power p plus 2 plus partial derivative with respect to z of negative b z over norm of r power p plus 2 okay if we take a the partial derivative of negative x over x squared plus y square plus z squared power p plus 2 over 2 the derivative with respect to x equal negative p over x squared plus y square plus z square power p plus 2 partial derivative with respect of x with respect to x plus negative p x differentiate partial derivative with respect to x of 1 over x squared plus y square plus z squared power p plus 2 over 2 okay which is equal to negative p over x square plus y square plus z square power p plus 2 over 2 minus p x multiply negative p plus 2 over 2 multiply 2x over x squared plus y squared plus z squared power p plus 4 over 2 which is equal to negative p over the norm of r power p plus 2 plus p multiply p plus 2 multiply x squared over the norm of r power p plus 4 similarly to y and z we get the dot product of the curve, multiply negative p r over the norm of r power p plus 2 equals negative p over the norm of r power p plus 2, multiply 3 plus p, multiply p plus 2, multiply x square plus y squared plus plus z squared over the norm of r power p plus 4...

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