A discrete form of the theorem that each field endomorphism of R(Qp)\mathbb{R}{\left( {\mathbb{Q}_{{\text{p}}} } \right)}R(Qp) is the identity (original) (raw)
Related papers
On the arithmetic of the endomorphisms ring
2012
For a prime number p, Bergman (1974) established that End(Z p × Z p 2) is a semilocal ring with p 5 elements that cannot be embedded in matrices over any commutative ring. We identify the elements of End(Z p × Z p 2) with elements in a new set, denoted by E p , of matrices of size 2 × 2, whose elements in the rst row belong to Z p and the elements in the second row belong to Z p 2 ; also, using the arithmetic in Z p and Z p 2 , we introduce the arithmetic in that ring and prove that the ring End(Z p × Z p 2) is isomorphic to the ring E p. Finally, we present a Die-Hellman key interchange protocol using some polynomial functions over E p dened by polynomial in Z[X]. x + y = (x + y) mod m and x • y = (xy) mod m, for all x, y ∈ Z m. Let us assume from now on that p is a prime number and consider the rings Z p and Z p 2. Clearly, we can also assume that Z p ⊆ Z p 2 , even though Z p is not a subring of Z p 2. Then, it follows that notation is utmost important to prevent errors like the following. Suppose that p = 5, then Z 5 = {0, 1, 2, 3, 4} and Z 5 2 = {0, 1, 2, 3,. .. , 23, 24}. Note that 2, 4 ∈ Z 5 and 2 + 4 = 1 ∈ Z 5 ; but 2, 4 ∈ Z 5 2 equally. However when 2, 4 ∈ Z 5 2 , 2 + 4 = 6 ∈ Z 5 2. Obviously, 1 = 6 in Z 5 2. Such error can be easily avoidable if we write, when necessary, x mod p and x mod p 2 to refer the element x when x ∈ Z p and x ∈ Z p 2 , respectively. In this light, the above example could be rewritten as (2 mod 5)+(4 mod 5) = 1 mod 5, whereas (2 mod 5 2) + (4 mod 5 2) = 6 mod 5 2. 2 The ring End(Z p × Z p 2) Consider the additive group Z p × Z p 2 of order p 3 , where the addition is dened componentwise, and the set End(Z p × Z p 2) of endomorphisms of such additive group. It is well known that End(Z p × Z p 2) is a noncommutative unitary ring with the usual addition and composition of endomorphisms, that are dened, for f, g ∈ End(Z p × Z p 2), as (f + g)(x, y) = f (x, y) + g(x, y) and (f • g)(x, y) = f (g(x, y)). The additive and multiplicative identities O and I are dened, obviously, by O(x, y) = (0, 0) and I(x, y) = (x, y) respectively. The additive identity is also called the null endomorphism. Te next result not only determines the cardinality of the ring End(Z p × Z p 2), but also introduces the primary property of such a ring: it cannot be embedded in matrices over any commutative ring. Theorem 1 (Theorem 3 of [2]) If p is a prime number, then the ring of endomorphisms End(Z p × Z p 2) has p 5 elements and is semilocal, but cannot be embedded in matrices over any commutative ring.
MATH 20201: ALGEBRAIC STRUCTURES I (SECTIONS 1-5
Definition. A binary operation * on a non-empty set S is a rule that assigns to each ordered pair of elements of elements of S a uniquely determined element of S. The element assigned to the ordered pair (a, b) with a, b ∈ S is denoted by a * b. Remark. In other words, a binary operarion of a set S is a function * : S×S → S from the Cartesian product S × S to the set S. The only difference is that the value of the function * at an ordered pair (a, b) is denoted by a * b rather than * ((a, b)). Examples. Let S = N = {1, 2, 3,. . .}. (1) a ⋆ b = max(a, b) e.g. 2 ⋆ 3 = 3, 3 ⋆ 2 = 3, 3 ⋆ 3 = 3. (2) a ⋄ b = a e.g. 2 ⋄ 3 = 2, 3 ⋄ 2 = 3, 3 ⋄ 3 = 3. (3) ab = a b e.g. 23 = 2 3 = 8, 32 = 3 2 = 9, 33 = 3 3 = 27.
Chapter 1. Polynomial automorphism groups 1. What is a polynomial automorphism? 2. Recognizing an automorphism 3. The case n = 1, R a domain 4. Notation 5. Canonical homomorphisms 6. Subgroups 7. Formal inverse 8. More subgroups 9. More on n = 1 10. Elementary and tame automorphisms 11. Stable tameness 12. The case n = 2, R a field 13. Structure of GA 2 (k), k a field 14. Structure of T 2 (R), R a domain 15. The case n = 3 Chapter 2. Perhaps the Jacobian Conjecture is simple IX X SUGGESTED NOTATION FOR GROUPS OF POLYNOMIAL AUTOMORPHISMS The general linear group. We then have the general linear group GL n (R) naturally contained in GA n (R) as the subgroup of all F = (F 1 ,. .. , F n) for which each F i is a linear form. Here an invertible matrix A is identified with the automorphism F given by the vector (A • X) t , where X = (X 1 ,. .. , X n) t. We can view GL n (R) anti-isomorphically as the stabilizer in R [n] of the Rmodule RX 1 ⊕ • • • ⊕ RX n. The origin preserving group. The group of "origin preserving" automorphisms, i.e., those F ∈ GA n (R) for which F (0) = 0, is denoted by GA 0 n (R). Note that GL n (R) ⊆ GA 0 n (R). The tangent preserving groups. The group of "tangent preserving" automorphisms, denoted by GA 1 n (R), consists of all F = (F 1 ,. .. , F n) for which F i has the form X i + terms of order ≥ 2. More generally, for k an integer ≥ 1 we have the subgroups GA k n (R) whose F have the property F i = X i + terms of order ≥ k + 1. We note that ∩ k GA k n (R) = {(X 1 ,. .. , X n)}. The affine group. The affine group consists of those automorphisms of the form F = (L 1 + c 1 ,. .. , L n + c n) where L 1 ,. .. , L n are linear forms in R [n] and c 1 ,. .. , c n ∈ R. We denote this group by Af n (R). Note that GL n (R) ⊆ Af n (R), and, in fact, GL n (R) = Af n (R) ∩ GA 0 n (R). The group of translations. Also contained in the affine group is the group of translations, i.e., those automorphisms of the form F = (X 1 + c 1 ,. .. , X n + c n) with c 1 ,. .. , c n ∈ R. We denote this subgroup by Tr n (R). It is isomorphic to the additive group R n. The elementary group. An automorphism is called elementary or de Jonquiere, if it has the form
Èíñòèòóò ïðîáëåì èíôîðìàòèêè è óïðàâëåíèÿ ÌÎÍ ÐÊ 480100 Àëìàòû, Ïóøêèíà óë., 125, vvv@ipic.kz Ïóñòü H ïëîòíàÿ ïîäãðóïïà â R. Ìû äîêàaeåì ÷òî òîãäà (R, <, +, 0, H, H q , e p,q,i ) q∈Q + äîïóñêàåò ýëèìèíàöèþ êâàíòîðîâ, ãäå H q = {q · h : h ∈ H} äëÿ íåêîòîðîãî q ∈ Q + . Åñëè äëÿ íåêîòîðûõ p, q ∈ Q + èìååò ìåñòî |H q :
A Commutativity Theorem For Associative Rings
1995
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.