#### Life after TRUTH: I. Determination — 3 comments

: One thing I don’t like about story-telling is them stories sitting still every time the teller goes … more →

#### Surjective Proof

: Proof Surjective (or Onto 满射 ) Let f: R² -> R² (x,y) -> (x+1, y+2) Prove f onto? Proof: By def … more →

Tags: Modern math

#### More about functions — 1 comment

: So let’s talk some more about different kinds of functions:there are Injective functions which … more →

#### Quotient Modules — 3 comments

: Point of Post: In this post we construct quotient modules and discuss some of the important theorems … more →

#### No Surjection to the Power Set

: Problem? Let be a set and is the set of all subsets of . Prove that there is no surjective function … more →

Tags: Set theory, function, power-set

#### 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 . W … more →

#### Finitely generated modules over R are precisely the module-homomorphic images of Rⁿ

: Let be a ring. Prove that is a finitely generated module over if and only if is a module-homomorphic … more →

#### Injectivity/surjectivity duality — 2 comments

: 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)

: 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) — 3 comments

: 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 — 2 comments

: 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

: 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 — 2 comments

: 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

: 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

: 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

: 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

: 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

: The surface of a sphere is painted with black and white paints such that the area of black paint is … more →

#### Lecture 11

: Today in the lecture we discussed three properties that linear transformations can have. They are: i … more →

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