A CLASS OP SIMPLE GAMMA DELTA RINGS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By CARL CHRISTOPHER MANERI, B. S -Yc <K- # -K- *- The Ohio State University 195>9 Approved by: Srvoiw>. kJ^uJUXct __________ Adviser Department of Mathematics ii I wish to express my appreciation to Professor Erwin Kleinfeld for his encouragement and guidance in the research for this dissertation. TABLE OP CONTENTS Introduction ........................................... 1 Section 1: Preliminary Discussion and Definitions .... 6 Section 2: General Identities ....................... 10 Section 3: Identities Involving anIdempotent ........ 18 Section ij.: Pierce Decomposition ..................... 2l± Section 5>i Further Results .......................... 33 Section 6: Appendix .................................. 39 Bibliography ........................................... k 2 Autobiography .......................................... I4I4- iil INTRODUCTION The study of non-associative algebras began when Cayley introduced the Cayley numbers In 18J4.5 rs) • Their generalization to the Cayley-DIckson algebras gives a class of simple alternative rings. It was found by E. Kleinfeld that except for characteristic 3, these algebras are the only simple alternative rings which are not associative. For characteristic 3 he showed that the Cayley-Dickson algebras are the only simple alternative rings which are not either associative or nil £7*8,93 • The study of simple rings was initiated in 1908 when Wedderburn classified all simple finite dimensional associative algebras as either trivial algebras with all products zero or total matric algebras over a division ring Ll6]J . Semi-simple associative algebras of finite dimension are the direct sura of a finite number of simple algebras. From the above examples one can see the importance of looking at simple rings in the study of any class of rings. Simple rings are one of the fundamental building blocks of general rings. A. A. Albert introduced the study of right alternative rings in [2] . He proved that a simple finite dimensional right alternative algebra of characteristic not 2 is alternative [3} • Skornyakov proved that a right alternative division ring of characteristic not 2 is alternative[l3Q . This was later generalized by Kleinfeld when he proved that a right alternative ring of characteristic not 2, with the property that (x,y,z) = 0 implies (x,y,z) = 0, is alternative [ 6"\ . R. L. San Soucio found an example of a right alternative division ring of characteristic 2 which is not alternative tiiQ • San Soucie gives the necessary and sufficient conditions for a right alternative division ring of characteristic 2 to be alternative r u ] - In 19ij-9 Albert introduced a new class of algebras which he called almost alternative algebras (their definition as well as others to be discussed will be given in Section 1). Within the class of almost alternative algebras are what Albert called the ( Y, % ) algebras. Albert proved in Cl] that practically all of the almost alternative algebras are equivalent to either right alternative algebras or to (Yj £ ) algebras ( a more precise form of this statement appears in Section 1). This equivalence preserves ideals, nilpotence of elements and therefore preserves nil ideals. Thus one can get information about the general almost alternative algebras from a study of ( Yj ^ ) algebras. Some of the results on ( YjS ) algebras and rings will now be discussed. Kokoris showed that a simple ( £ ) algebra with £ ^ 0,1 and characteristic not 2, 3 or 5> is either associative or possesses an absolutely primitive unity element r i a l . Kleinfeld showed that an idempotent of a simple ( £) ring R, with i f 0,1, which is of characteristic not 2 or 3 and which is not associative is the unity element of R. This appears as the Appendix of [ 13 *) . Kokoris in proved that an idempotent e in a simple (1,1) ring R of characteristic not 2 is either the unity of R or R is associative. In a paper on (1,1) algebras i ih , Kleinfeld showed that simple (1,1) rings which are not associative and have characteristic not 2 or 3 possess no proper left ideals C ul . He also showed in this paper that a (1,1) ring of characteristic not 2 or 3 which has no proper right ideals is associative. His main result in this paper is that simple (1,1) algebras of characteristic not 2, 3 t or 5 are associative. In this same paper Kleinfeld showed that a ( V> £ ) ring R, with £ ^ 0,1, possesses no proper left or right ideals if R is simple and of characteristic not 2 or 3 and is not associative. The principal result in this dissertation is that an idempotent e of a simple (-1,1) ring R is the unity element of R if R is simple, not associative and of characteristic not 2 or 3. This eliminates one class of simple (-1,1) rings. One might conjecture that all simple (-1,1) rings (with the properly chosen characteristic) are associative. We shall also show that, if R is a simple not associative (-1,1) ring of characteristic not 2 or 3, then R possesses no proper left ideals. The attempt to do the same for right ideals was unsuccessful. The results of Albert on right alternative algebras tell us that a simple (-1,1) algebra of characteristic not 2 or 3 is associative (this is discussed in Section 1). These results can be summarized as follows: 1. Any simple ( Vj % ) ring which possesses an idempotent is either associative or that idempotent is the unity of the ring. 2. Any simple ( £ ) algebra with $ = 0,1 is associative. 3- Any simple (Vj &) ring cannot contain proper left ideals and proper right ideals at the same time, unless it is associative. If & ^ 0,1 it can contain neither, unless it is associative. Section 1 gives background material and definitions. In Section 2 we derive some identities in an arbitrary (-1,1) ring. Section 3 brings in the Idempotent and simplicity. Section ij. on the Pierce decomposition contains the main result. Section 5 is the result on left Ideals mentioned above. Section 6 contains an example of a (-1,1) algebra which possesses an Idempotent not the unity. 5 The entire dissertation is self contained except for the proof of identity (28). This requires two lemmas from Kleinfeld1s paper on right alternative rings 16^ . * 1. PRELIMINARY DISCUSSION AND DEFINITIONS Since we are dealing with, non-associative rings, we define the associator of x,y,z, where x,y,z are elements of our ring, to be (x,y,z) = (xy)z - x(yz). We also define the commutator to be (x,y) = xy - yx. The alternative laws are (x,x,y) « 0 and (y,x,x) = 0. The first is the left alternative law while the second is the right alternative law. An alternative ring is one which satisfies both of these laws. The flexible law is (x,y,x) = 0. A ring or algebra is defined to be of type ( Yj £ ) if the following identities hold: (1) A(x,y, z) sa (x,y, z) + (y,z,x) + (z,x,y) = 0, (2) (z,x,y) + Y(x,z,y) + S(y,z,x) = 0, with y 2 - S2 + £ = 1. Albert defined an almost left alternative algebra to be a finite dimensional algebra A over a field F of characteristic not 2 with the following properties: I. For fixed T-,T\ * F 6 7 (3) z(xy) = o((zx)y + Q (zy)x + V ( xz)y + &(yz)x + 6 y( zx) +t^x(zy) + 0-y(xz) +Tx(yz), for any x,y,z in A. II. (x,x,x) = 0 for x ^ A . III. There exists an algebra B over F such that B is not commutative, possesses a unity element and satisfies identity (3). An algebra A over F is called almost right alternative if I, II and III are valid in A but with identity (3) replaced by an identity of the same form except that z(xy) is replaced by (xy)z. An algebra A over F is called almost alternative if it is both almost left alternative and almost right alternative. For algebras over a given field F, of characteristic not 2, an equivalence relation is defined in the following way. Let A be an algebra over F. Let B be the algebra over F consisting of the same vector space as A but with a different multiplication. In other words A and B are identical as vector spaces. Let x,y£A. Then xy will denote the product in A and x*y will denote the product in B. A and B are called quasiequivalent if the identity x*y = ^xy + (l- *X)yx holds for arbitrary x , y 6 A and with a fixed > 6 F where X t i. Albert shows in E 1} that if A is an almost alternative algebra which is not flexible with vl + P / 3/Kt then there is a finite extension K of P such that the algebra ak , where AK is the algebra over K obtained by enlarging the base field from P to K, is quasiequivalent with ^ ^ K to an algebra B where B is a left alternative algebra or is a ( & ) algebra. We can note very easily that ideals are preserved by quasiequivalence as well as nilpotence of an element. Hence nil ideals are preserved. In a (-1,1) ring (2) reduces to (z,x,y) - (x,z,y) + (y,z,x) = 0. If this is subtracted from (1) we get (i+) B(x,y,z) = (x,y,z) + (x,z,y) = 0. If we set y = x in (z,x,y) - (x,z,y) + (y,z,x) = 0, we get (5) (Z,x,x) = 0.