Finite Fields
... If φ is a ring homomorphism from a ring R onto a ring S then the factor ring R/kerφ and the ring S are isomorphic by the map r + kerφ 7→ φ(r). We can use mappings to transfer a structure from an algebraic system to a set without structure. Given a ring R, a set S and a bijective map φ : R → S, we ca ...
... If φ is a ring homomorphism from a ring R onto a ring S then the factor ring R/kerφ and the ring S are isomorphic by the map r + kerφ 7→ φ(r). We can use mappings to transfer a structure from an algebraic system to a set without structure. Given a ring R, a set S and a bijective map φ : R → S, we ca ...
Brauer-Thrall for totally reflexive modules
... (1.2) Theorem. If there exists a totally reflexive R-module without free summands, which is presented by a matrix that has a column or a row with only one non-zero entry, then that entry is an exact zero divisor in R. These two results—the latter of which is distilled from Theorem (5.3)—show that ex ...
... (1.2) Theorem. If there exists a totally reflexive R-module without free summands, which is presented by a matrix that has a column or a row with only one non-zero entry, then that entry is an exact zero divisor in R. These two results—the latter of which is distilled from Theorem (5.3)—show that ex ...
Polynomial Rings
... Just as in the case of the integers, each use of the Division Algorithm does not change the greatest common divisor. So the last pair has the same greatest common divisor as the first pair — but the last pair consists of 0 and the last nonzero remainder, so the last nonzero remainder is the greates ...
... Just as in the case of the integers, each use of the Division Algorithm does not change the greatest common divisor. So the last pair has the same greatest common divisor as the first pair — but the last pair consists of 0 and the last nonzero remainder, so the last nonzero remainder is the greates ...
4. Morphisms
... Exercise 4.13. Let X ⊂ A2 be the zero locus of a single polynomial ∑i+ j≤d ai, j x1i x2j of degree at most d. Show that: (a) Any line in A2 (i.e. any zero locus of a single polynomial of degree 1) not contained in X intersects X in at most d points. (b) Any affine conic (as in Exercise 4.12 over a f ...
... Exercise 4.13. Let X ⊂ A2 be the zero locus of a single polynomial ∑i+ j≤d ai, j x1i x2j of degree at most d. Show that: (a) Any line in A2 (i.e. any zero locus of a single polynomial of degree 1) not contained in X intersects X in at most d points. (b) Any affine conic (as in Exercise 4.12 over a f ...
