Blogs about: Surjective
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Characterize the elements of a semigroup of transformations on a set which generate trivial left and right principal ideals
Let be a set and let denote the semigroup of functions . Characterize the elements such that and . We begin with some lemmas about functions. Recall that a function is called injective if for all , if… more →
Project Crazy Project
Life after TRUTH: I. Determination
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wrote 1 month ago: One thing I don’t like about story-telling is them stories sitting still every time the teller goes … more →
Surjective Proof
wrote 1 month ago: Proof Surjective (or Onto 满射 ) Let f: R² -> R² (x,y) -> (x+1, y+2) Prove f onto? Proof: By def … more →
More about functions
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wrote 3 months ago: So let’s talk some more about different kinds of functions:there are Injective functions which … more →
Quotient Modules
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wrote 1 year ago: Point of Post: In this post we construct quotient modules and discuss some of the important theorems … more →
No Surjection to the Power Set
wrote 1 year ago: Problem? Let be a set and is the set of all subsets of . Prove that there is no surjective function … more →
Characterize the elements of a semigroup of transformations on a set which generate trivial left and right principal ideals
wrote 1 year ago: Let be a set and let denote the semigroup of functions . Characterize the elements such that and . W … more →
Finitely generated modules over R are precisely the module-homomorphic images of Rⁿ
wrote 1 year ago: Let be a ring. Prove that is a finitely generated module over if and only if is a module-homomorphic … more →
Injectivity/surjectivity duality
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wrote 1 year ago: Yet again I’m unable to sleep, so here’s something else I was thinking about. Suppose we … more →
Quotient Maps and Saturated Sets (III)
wrote 1 year ago: I just want to mention something briefly that I forgot to in the last post. Lemma: Let be a quotient … more →
Quotient Maps and Saturated Sets (II)
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wrote 1 year ago: As usual in mathematics, there are several different characterisations of objects, and saturated set … more →
If I is nilpotent and an induced R-module homomorphism M/IM → N/IN is surjective, then M → N is surjective
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wrote 2 years ago: Let be a commutative ring and let be a nilpotent ideal, say with . Let and be left -modules and supp … more →
If n divides m, then the natural ring homomorphism from ZZ/(m) to ZZ/(n) is surjective on the units
wrote 2 years ago: Let and be positive integers with dividing . Prove that the natural surjective ring homomorphism is … more →
Find all the ring homomorphic images of the integers
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wrote 2 years ago: Find all homomorphic images of . Recall that every additive subgroup of is cyclic, and moreover the … more →
On finite groups, power maps with exponent relatively prime to the group order are surjective
wrote 3 years ago: Let be a cyclic group of order and let be an integer relatively prime to . Prove that the map is sur … more →
Surjective group endomorphisms need not be automorphisms
wrote 3 years ago: Let . Prove that for any fixed integer the map given by is a surjective group homomorphism but is no … more →
Surjective group homomorphisms preserve abelianicity
wrote 3 years ago: If is an isomorphism, prove that is abelian if and only if is abelian. If is only a homomorphism, wh … more →
Demonstrate that a given relation defined in terms of a surjective function is an equivalence relation
wrote 3 years ago: Let be a surjective map of sets. Prove that the relation on given by iff is an equivalence relation … more →
Black and white painted sphere
wrote 3 years ago: The surface of a sphere is painted with black and white paints such that the area of black paint is … more →
Lecture 11
wrote 4 years ago: Today in the lecture we discussed three properties that linear transformations can have. They are: i … more →
