Semigroup rings over semiprime ring semigroups (Developments of Language, Logic, Algebraic system and Computer Science)

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(1)96. 数理解析研究所講究録第2051巻 2017年 96-100. Semigroup rings. over. semiprime ring semigroups. Hisaya Tsutsui Department of Mathematics Embry‐Riddle Aeronautical University Prescott, AZ USA E‐mail address: Hisava.Tsutsuii,erau.edu. 1. Introduction. The talk. presented was based. (ERAU). More detailed LetR be. a. semigroup areas. R[S] is. a. let S be. ring, and structure. of both. and. on‐goingjoint work with Y.Hirano (Naruto) and. completed version ofthis manuscript will. a. \mathrm{o}\mathrm{f}S and the. with. a. ring. find that the structure. structure. in. an. object of great utility. More. every normal finite. Consequently,. one. prime. if and. ,. characterized. G|. only. ,. by the. then K[G] is. \mathrm{i}\mathrm{f}R is. a. \mathrm{o}\mathrm{f}G is. a. an. primeness. arbitrary semigroup.. structure. semigroup, then R[\mathrm{S}] is semiprime. or. This is. progress has been made in recent years. For. again possible to determine the. ,. instance, much a. group.. Here,. is we. \mathrm{i}\mathrm{f}R is. a. field. Furlher variations include nontrivial finite normal. semiprime. and the order of. non‐zero‐divisor in R[5].. may ask whether the. where S is. only. \mathrm{o}\mathrm{f}R,S and. finite group and K is. a. semisimple [3].. ifand. in various. structure \mathrm{o}\mathrm{f}R and various finiteness. prime ring and G has li0. generally, R[G] is semiprime subgroup. structures. full of fascinating results. For. .. subgroup [1].. some. resulting. \mathrm{o}\mathrm{f}G Maschkes famous theorem states that \mathrm{i}\mathrm{f}G is. the result that R[G] is. be submited elsewhere.. \mathrm{o}\mathrm{f}R[G] where G is notjust a semigroup but. whose characteristic does not divide |. case. structure. \mathrm{o}\mathrm{f}R[G] is exactly. B. Solie. semigroup ring R[S] simultaneously encodes the. long mathematical history. already known about the. in the. The. semigroup.. ring theory and group theory. The interplay between the. topic. properties. on an. \mathrm{o}\mathrm{f}R[S]. .. whenever R is. semiprimeness \mathrm{o}\mathrm{f}R[S] can be characterized. a. much. highly For. more. difficult. question. on. which. structured classes of semigroups, it is. example,. when S is. semiprime [4].. a. When F is. cancellative a. field and S is. finite,.

(2) 97. a. version of Maschkes theorem shows that the. nonsingularity We consider. arise. 2. as. the. of the structure matrix for S when viewed. semigroup rings over a particular class. multiplicative semigroup. Main Results and. Let K be. a. field and letS be. the set of all. a. K[R] is. only. not. ,. we. may. if every proper. Ann(K[0])contains K[0] LetR be. Theorem 1 is. semiprime. The. semigroup.. a. a. ring R. ,. we. and letK be. a. semigroup ringK[S] consists. for all s,t\in. ideal. \mathrm{o}\mathrm{f}K[R]. is. a. .. .. of. equip K[S] with the. We. S.. Note that. \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}K[R] is. a. direct. prime. Every prime. \mathrm{o}\mathrm{f}K[R] contained. 0. \mathrm{a}\mathrm{s}K[R]=K[0]\oplus Ann(K[0]). sum. ideal. oftwo. simple rings. ,. if and. \mathrm{o}\mathrm{f}K[R] except. in \mathrm{A}nn(K[0]) does not containK [0].. ring and K be a f $\iota$ eld ofcharacteristic zero. IfK[R] is semiprime, then R holds. ifR is a. commutative. ring or a domain.. ring and letK be afield ofcharacteristic zero. Then K[R] is. a. local ring. by J(R). a. .. withfimitely many units, letR^{*} denote. field of characteristic zero.. is. finite semisimple ring.. shall denote the Jacobson radical \mathrm{o}\mathrm{f}R. LetR be. Theorem 3. semigroups which. forget addition and thereby obtain a semigroup (R,\cdot) having both. semisimple Artinian ring ifand only ifR. For. Recall that the. s\hat{t}\wedge= \hat{st}. where. and every ideal. converse. LetR be. Theorem 2. a. ,. F[2].. ring.. It is easy to observe. nonzero. over. of semigroups: those. \mathrm{b}\mathrm{y}K[R]\mathrm{t}\mathrm{h}\mathrm{e} semigroup \mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}K[(R,\cdot)]. prime ring.. a. matrix. by the. Examples. multiplication,. ring (R,+,\cdot). and ]. We denote. as a. \displaystyle \sum_{s\in \mathcal{S} k_{s}\hat{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}k_{s}=0 for all but finitely many s\in S. sums. usual addition and Given. a. of a. characterized. semisimplicity \mathrm{o}\mathrm{f}R[S] is. Then. the group. J(K[R])=\displaystyle \sum_{r\in/(R)}K ( \hat{r} ‐Ô) and. of units. in R. K[R]lJ(K[R])\equiv K[0]\oplus K[R^{*}]. LetK be. an. algebraically closed field. of characteristic zero, and. modulo n It is immediate that \mathb {Z}_{n} is. semiprime. following proposition. of Theorem 1.. .. as a. corollary. if and. only. ifn is. 1\mathrm{e}\mathrm{t}\mathb {Z}_{n} denote the ring of integers squarefree,. and thus. we. have the.

(3) 98. LetK be. Proposition. an. algebraically closedfield ofcharacteristic zero,. and let\mathbb{Z} denote the .. ring ofintegers modulo n Then K[\mathbb{Z}_{n}] is semiprime ifand only ifn is squarefree. .. We. now. present 1. Example. a. few. examples. It is clear that. of the structure. \mathrm{o}\mathrm{f}K[R] for. finite. some. ring R.. K[\mathbb{Z}_{2}]\cong K\oplus K.. Let V be. a vector space over K We define a multiplication on the K ‐linear space K\oplus V by the formula (a,v)\cdot(b,w)=(ab,aw+bv) for any a, b\in K, v, w\in V Then K\oplus V becomes a K ‐algebra, .. .. which. denote. we. Example 2 two sided. byKW.. Consider the. ringK[\mathbb{Z}_{4}] and let g_{i}= î−Ôfori =1,2,3. idealsK[0] andS=Kg_{1}+Kg_{2}+Kg_{3}. e_{1}=\displaystyle \frac{1}{2}(g_{1}-g_{3}) e_{2}=\displaystyle \frac{1}{2}(g_{1}+g_{3}) ,. andgl =e_{1}+e_{2} Example sum. .. We. can. easily. 3 Consider the. .. see. The. .. Then e_{1}, e_{2}are. identity. Then K[\mathbb{Z}_{4}] is the direct sum. .. of the. ringS. us. of. set. S. .. ringK[\mathbb{Z}_{6}] and let g_{i}= î‐Ôfori =1,2,\cdots,5 .. .. orthogonal central primitive idempotents of. \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}K[\mathbb{Z}_{4}]\equiv K^{2}\oplus(K\mathfrak{B}\mathrm{i}). ofK[0] andS=K\mathrm{g}_{1}+Kg_{2}+\cdots+Kg_{5} Setel. is g_{1} Let. =\displaystyle \frac{1}{2}(g_{1}+g_{5}). Then K[\mathbb{Z}_{6}] is the direct. .. 2=\displaystyle \frac{1}{2}(g_{1}-g_{S}). ande. .. We. again have. orthogonal central primitive idempotents e_{1} ande2 in S andgl =e_{1}+e_{2} and moreoverelS \equiv K^{3} ,. ande2S\equiv K^{2}. Example us. Thus. we. thatK[\mathbb{Z}_{6}]\cong K^{6}. have. 4 Consider the. direct sum Let. .. ring K[\mathbb{Z}_{8}] and let. .. The. the. identity ofthe ringS is g_{1}.. set. Then e_{1}, e_{2}, e_{3}, e_{4}. that. for i=1,2,\cdots,7. ThenK[\mathbb{Z}_{8}] is. oftwo sided ideals K[Ô] andS=Kg_{1}+Kg_{2}+\cdots+K\mathrm{g}_{7}. e_{1}=\displaystyle \frac{1}{4}(g_{1}-g_{3}-g_{5}+g_{7}) e_{2}=\displaystyle \frac{1}{4}(g_{1}+g_{3}-g_{5}-g_{7}) e_{3}=\displaystyle \frac{1}{4}(g_{1}-g_{3}+g_{5}-g_{7}) e_{4}=\displaystyle \frac{1}{4}(g_{1}+g_{3}+g_{5}+g_{7}). We. g_{i}=\^{i}-\^{O}. can. easily. see. are. orthogonal. e_{1}S\cong K. ,. .. central. e_{2}S\cong K. ,. primitive idempotents. e_{3}S\equiv K\triangleright K. K[\mathbb{Z}_{8}]\equiv K^{3}\oplus(K\triangleright K)\oplus(K\ltimes(K\oplus K)). .. ,. and. of S and g_{1}=e_{1}+e_{2}+e_{3}+e_{4}.. e_{4}S\cong K\ltimes(K\oplus K) .Therefore. we. have.

(4) 99. R=\{[_{0}^{a}0 ab0 dac)|a, b, c, d\in GF(2)\} R^{*}=\{ left\{ begin{ar y}{l } a&b&c\ 0&a&d\ 0&0&a \end{ar y}\right\}|a\neq0,b c,d\inGF(2)\}. Example. Then. 5. Consider the ring. .. K[R^{*}]\cong K^{4}\oplus M_{2}(K) K[R]lJ(K[R])\equiv K^{5}\oplus M_{2}(K) D_{8}. oforder 8 and. so. .. We. can. easily. see. oforder |R|=2^{4}=16.. \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}R^{*} is the dihedral. Therefore, by Theorem 3,. we. group. have that. .. LetM_{2}(GF(2)) denote the ring of 2\times 2 matrices over the field GF(2) Then we Z[M_{2}(GF(2))] is a semiprime ring. Let us setH=M_{2}(GF(2))-GI_{2}(GF(2)) we can see that Q[H]\cong Q\oplus M_{3}(Q) In fact, let. Example. ó. .. canprove that. Then. .. .. o=\left(\begin{ar y}{l 0&0\ 0&0 \end{ar y}\right),e_{1}=\left(\begin{ar y}{l 1&0\ 0&0 \end{ar y}\right),e_{2}=\left(\begin{ar y}{l 0&0\ 0&1 \end{ar y}\right),e_{3}=\left(\begin{ar y}{l 0&\mathrm{l}\ 0&0 \end{ar y}\right), e_{4}=\left(\begin{ar y}{l 0&0\ \mathrm{l}&0 \end{ar y}\right),e_{5}=\left(\begin{ar y}{l \mathrm{l}&0\ 1&0 \end{ar y}\right),e_{6}=\left(\begin{ar y}{l 0&1\ 0&1 \end{ar y}\right),e_{7}=\left(\begin{ar y}{l 1&\mathrm{l}\ 0&0 \end{ar y}\right), e_{8}=\left(\begin{ar y}{l 0& \ 1& \end{ar y}\right),e_{9}=\left(\begin{ar y}{l \mathrm{l}&\mathrm{l}\ \mathrm{l}&1 \end{ar y}\right). Then the elements E. =. Ô, F_{1}=\hat{e}_{1} ‐Ô, F_{2}=\hat{e}_{2}-\hat{O},. F_{3}=(\hat{e}_{1}-\hat{O})+(\hat{e}_{2}-\hat{O})+(\hat{e}_{3}-\hat{O})+(\hat{e}_{4}-\hat{O})-(\hat{e}_{5}-\hat{O}) -(\hat{e}_{6}-\hat{O})-(\hat{e}_{7}-\hat{O})-(\hat{e}_{8}-\hat{O})+(\hat{e}_{9}-\hat{O}) are. primitive orthogonal idempotents, and Q[H]=QE\oplus Q[H](F_{1}+F_{2}+F_{3})\equiv Q\oplus M_{3}(Q). Since GL_{2}(GF(2))\equiv S_{3} It is. easily. see. ,. we. have. Q[M_{2}(GF(2))]lQ[H]\equiv Q[S_{3}].. that Q[S_{3}]\equiv Q\oplus Q\oplus M_{2}(Q) Hence Q[M_{2}(GF(2))]\mathrm{i}\mathrm{s} .. semisimple Artinian ring Q\oplus Q\oplus Q\oplus M_{2}(Q)\oplus M_{3}(Q). .. isomorphic to. the. ..

(5) 100. References [1]. Ian G. Conmell. On the group. [2] Michael P.. ring. Canad.. J.. Math., 15:650‐685, 1963.. Drazin. Maschkes theorem for semigroups. J.. Algebra,72(10): 269‐278,. 1981.. [3]. T. Y. Lam. A frst. course. Texts in Mathematics.. in noncommutative. rings, volume 131 of Graduate. Springer‐Verlag, New York, second edition,. [4] Jan Okninski. Prime ideals ofcancellative semigroups. 32(7):2733-2742 2004.. Comm.. 2001.. Algebra,. ,. [5]. Donald S. Passman.. 1971. Pure and. Inf_{2}ite group rings. Applied Mathematics, 6.. Marcel Dekker, Inc., New York,.

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