设 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 是一个交换环. [tex=2.929x1.357]wNZYyFzs1adFMqSe7mkJQQ==[/tex] 表示 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 上的全体 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶方阵的集合. 与数域的情况一样可定义[tex=2.929x1.357]wNZYyFzs1adFMqSe7mkJQQ==[/tex] 中的矩阵的加法和乘法运算. 证明: [tex=2.929x1.357]wNZYyFzs1adFMqSe7mkJQQ==[/tex] 关于这两种运算构成一个环.
证明 (1) 对任意的[tex=31.643x5.357]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[/tex]规定[p=align:center][tex=22.429x5.357]4BNHnJf9vindIrhIPRFaGV0Q7ODKf95M0G7An9EvV5/eoZyt7cN+vXllc8qOGkjaxxISrHXhcRvDwWB+z+Lvl/OlCpomIhzayDgFMAzc7apdHAzLwtI43hNl+bZoRqGqFkiETx6RPLHsfpAlo7jK0jUVt6oKx+gJ6zT1xU4hg+w49GXcv4M48jnOq+EQ+1RHLDqeClRW428QuNZk3kP0xLMM9jOY5u11JUra3kRc88ZpR8seB9cGoa2Oztcs0OKtAl8iLcJvV11NfVWyaJguWFD9CasKqEmPAHSlg/VnroGwPDAk31hfJGHlKa/FYwEdQLydPgegbPlJfEUQFcieOUtQ3IbdbZrMfC5S+QSZTZ8=[/tex][p=align:center][tex=21.286x4.5]OgVntXYK4O34noTl54/1w920YFXj5gq9gA6hWTc6fTGjYf9s7Hj9pQN8F+SYy7CnqoT+oynATnwqhYCJObPG0NWzRQl7rs/buguCqiDK3Ysa5ul91wwUH8LCNOOrmHHXpg082XlzOYBWZWhJZ0cSAObXU4CfskHiQsYINGlhJPoakEb2Llb0lHlsDItRVMziF4tDeW1W/gTdszTdigT1Q0O/x104D9qW3+cwrEU1Ov84VO++IyJ7TqNch/auJCosfeoawflcliKLHSiFfccTRRTBeImtv1odYs7gnqZ8vr/hd+HuRZ3AQrU9OPgdA77o[/tex]显然, 如此规定的加法与乘法是 [tex=2.929x1.357]wNZYyFzs1adFMqSe7mkJQQ==[/tex] 的代数运算.以下,[tex=2.071x1.357]29p00RMRAu0gcJ297UlgKA==[/tex] 表示矩阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 的第 [tex=0.357x1.0]O88k7AtkDgTC9kv/8dY0lg==[/tex] 行 [tex=0.429x1.214]rmIPPJrP+tFN2kAYPlU/4g==[/tex]列位置上的元素. (2) 对任意的[p=align:center][tex=31.643x5.357]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[/tex][p=align:center][tex=18.571x1.357]pIKh9Z7N8AffBV36SxvCF87+ix7U6bbvFEyqNgfnMpnqzOqo6kVulBBtHQrjJEvBqOG7so4syoiSKo+IFRuzZw==[/tex]于是 [tex=7.143x1.214]+LP5a91vF6ST23wGwYy8dQ==[/tex] 所以加法交换律成立.(3) 对任意的[tex=27.5x5.357]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[/tex][tex=16.929x4.5]0g5vxaagmyEBQHxTGP3jN4YPUykH5rNpnV+N5/qy867vP4ryzwFKVJPIqqgMRq0lOwTaLwPzc0+lsemx2SfDlv5uV4kVjhe6biFn0Yj/LSNpxcS8XhNlYWDvKmJ2k66lv+Ac8MeF1iF9X1VSVhkbnrVeernxch0WjmSS07JUhehWbG+wyiGqoiw5kev6dwVUBFUn7ViOOgtEahpJyPi5iBZjISjoEfL4Dmt/6qszQz4J1V+8u5U4oK5fYEBMbs2TCSikyhkVrQS1OsRPWBrfeQ==[/tex][tex=14.357x1.357]naTM6gCWFw6g52ms1cLnKiUOvAzTxMLrdVEMoi/VTrnqOFnCGZs0Rk487NqgOnhLysPzehv4aEq0R4PFDh6pxA==[/tex] [tex=7.857x2.857]kTlOlx4xpX+79smOY8Uo2zr7nrpW9Mg0UVsMngEm1KWsT98ixdgwIvWL/qZkSsZGh4hGprFo70GJ/BOgxXhvtA==[/tex]于是[tex=12.571x1.357]zLnnckaoPr4FMO0ZkIzLIk2SHw7JuRQkg85bBXmnGzU=[/tex]所以加法结合律成立.(4) 零矩阵 [tex=14.857x4.643]oBirXOOQkryNAsCSs0Xn5cH4Dc0zdACGlIEOTZQV/LPIOk14PpcGSSCN0W6io8X6mlJcO153Aii90QxpR/CqLuBQzGuHGQLlU0AxH1qKSnf8NPr/F+gT0Z+qyOefzz8X+qeaM9RQezsx5U3pGRNV/TrBJIHpfq/RngHm1vc0g0NVqyJCYuFP1KOzXBpr/heX2xkOpoAKXj9jXQC3fPRCLQ==[/tex]对任意的 [tex=5.214x1.357]WiG5/5e7sPAxAiK9li5GhYW7LDgefC+3NKOrUm0EdT0=[/tex] 有[p=align:center][tex=17.5x1.357]81UMzjzQYGQz4Kj4II5SWQhHmTkQDahUocRqeeIw7YvD9RV2eQCjOpFZes7fUApo9WqerGDCZWEslMHerf7/sw==[/tex]于是 [tex=8.643x1.214]buCWWytvU/VlK10pxB/+sQ==[/tex] 所以零矩阵为加法的零元.(5) 对任意的 [tex=17.571x4.5]3BT1BgBZQ5uJXxD5dg+w2ycQltJmLTacNGaiu1/xAheew5vSzZshzVC+1ir3xZiLyqjhrG7jLlbSn21wvD/VLZyuYCKyagvaHih1Vfqn7IlL3WcSVI7tnPH7YPCuZkWZRBn3s23wr2A53fnM4quqCw0C9RtaB1zMV0XFjsFrOx5fl3n4VetKymlhW9o0VwJhGp15wzNz48dx2C2K4uZstMMfFO/MTCFl7RSTl9jEVHRwGCBd2I3LS5Us4/hk3mzZqca9cbViI6ovWdlF2q+UqQ==[/tex]有[p=align:center][tex=20.357x5.214]JCgyfsV0+5j28mECzOtC22NnicY0kgUApozi1MtbdZFMUJ3pJM+8YB0z37yuvYVIASAwZmCO7RcyTF7pxZ3FDjvFgDeW/GFIXET4BkvLS+YQagkZrdnd+lB4CsgF/gAUqt72NvGnHuxpz7tX0Kztln6nKyBcdmr/c6Lg9Vp0Jl5aFtGbBmovwFL3Cz2suPGNLujwaMgQFT6JEQ/O23dyOlooi2qPj9XviNB3kQZibmCGt0PUJWAPiIP3JMQjE1pl9VMINuY6o7Uk/DOaFK4nW5dwQzfXleENUNpUDHE+j74=[/tex]且显然有[p=align:center][tex=22.929x1.357]GA9Atbvh180uZ81JGHIQ7t7e6dvwWwk5noaDKsYexYXaj3E6LDVa/cgDrBwUSEctgjqrkwMI0gEyD0ceyx0zJA==[/tex]于是 [tex=12.0x1.357]6WI08icEX/PWbT+LAR7knWflMr6KswPBfkKmamDxGkI=[/tex]所以 [tex=2.929x1.357]wNZYyFzs1adFMqSe7mkJQQ==[/tex]中每个元素都有负元.(6) 对任意的[tex=27.929x5.357]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[/tex][tex=16.929x4.5]0g5vxaagmyEBQHxTGP3jN4YPUykH5rNpnV+N5/qy867vP4ryzwFKVJPIqqgMRq0lOwTaLwPzc0+lsemx2SfDlv5uV4kVjhe6biFn0Yj/LSNpxcS8XhNlYWDvKmJ2k66lv+Ac8MeF1iF9X1VSVhkbnrVeernxch0WjmSS07JUhehWbG+wyiGqoiw5kev6dwVUBFUn7ViOOgtEahpJyPi5iBZjISjoEfL4Dmt/6qszQz4J1V+8u5U4oK5fYEBMbs2TCSikyhkVrQS1OsRPWBrfeQ==[/tex][tex=13.714x3.357]ZtVh5tf4eNwHTCnuTNKvbdzZwaa8GOAiYyTpYeeSisnC2duTIoq2iNfCTGLC/P7jvmpaTdg58oEzzMgUoUO7fsUnWB14x6gM9Zni8cRBHE19khPOe8ZQJr64eDjDYpyM[/tex] [tex=9.143x4.786]ndH7npQxGjXuZ3YmdlLp0p4F30+WCesTaC9bnQSkv9czHBDIs0qcldxU44oMiAfooX5VsexU94OgY+ctVjbUz1jB3V2S+XGaN7o7EhmLeho6v2VshW8vLr65t64DtRzQ[/tex]于是 [tex=7.643x1.357]xu5gUVDGweTV7hsrczZYIiOOIrYG90zBoLDhzVktDXk=[/tex] 所以乘法结合律成立.(7) 对任意的[tex=27.5x5.357]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[/tex][tex=16.571x4.5]XbmfowmH+x2njRlGmJod4ssHWzZfCc9JW0l68lpIEo3aJRC4w/ZPHqR3cs54QU+Fiil8FB8C+RWZrZ4vzL9IDBwn/dqecalxm8GLm6lnu8pssl/+TqGAqzEiVyQBhZaBcTJ9g0q/n4SlBUQSW1nztxkq7uh/syw7l9YJlGaUeo1DbGjMR+r3Ee0hrQd5CxCKWRJYzZ1GoOhxoBrsMIe0kQ8rXXsw6AbFt3o0DwhxvYQi38npp2fIrNl9/22h3Lv06VXk+bEFWyVEUh8M0KJ8kA==[/tex][tex=13.929x3.286]GwN/8uEmMGsbDmaWWVfNQlA8VGjwtnBkpcsnSyvQ+2yV/8LSbWobYVWtglE+sa1Kpa8R/fPHLWXaV1SSXDKzJgU2t9Hs8H4caVbadjkaJtU=[/tex] [tex=10.857x5.071]Ck4j1YFlvVH5wCAykOEMi3m4/TKSxmS4pv0p+hdfCOBQ7ys4PDFEcFEvpOQzi4g/RLV9akK8nw+KsHuHXisq9ALnx2pvXkRckvAifNyEhHPnx0UEcUWKO30iY/9jgLqorACRv8RMLHL9VYTxpJPp9Nt8QeQW78CHy7NQFJpYuvY=[/tex][tex=14.0x3.286]cIq8G6fYdz3JaCRlSVGqVCo/HBsVomldIA7FQcgCyddGwT03m00VMb17IOUb2Z2ibibxQHDYxsoZVGBJTFotoGipbnLnB9w6pHtSYaPoFlk=[/tex] [tex=10.857x5.071]Ck4j1YFlvVH5wCAykOEMi3m4/TKSxmS4pv0p+hdfCOAaVKvM4yUaCdQ/tdH1LcPSdsq6stijQud9FW2wd8X93lvAb3P3IMwkP7cvB0Yhlhjrfq+t+ymaregsMwBUJpGqcpn/rM4G27M2HyeDZ0sQDZGhuqbHYJDAEpbp5n1HjNc=[/tex] 于是 [tex=20.214x1.357]BK/UVoK6qXJV136ngyzkscUPUV7N9PkGCJ3NVf1jiPy4+9Abne2bXMkUzIf1S+6Q[/tex]所以乘法对加法的两个分配律成立.因此 [tex=2.929x1.357]wNZYyFzs1adFMqSe7mkJQQ==[/tex] 关于矩阵的这两种运算构成一个环.[p=align:center]
举一反三
- 设[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]是有单位元1的交换环，[tex=2.929x1.357]wNZYyFzs1adFMqSe7mkJQQ==[/tex]是[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]上[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶全阵环. [tex=5.714x1.357]b6x3UNsP/4RKtwWNsu9AsdyZ1gShonJONjXXZHDCNbo=[/tex] 证明[p=align:center][tex=9.714x1.286]yH0ulBrKXaGdnyIfiicOUMjyicceCdry3FxYqsokbx/jtaCNvHc1Md8nI7J0Mbfh[/tex]其中[tex=0.786x1.0]I/kNMtd8YcgkWCrgriW/hA==[/tex]是[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶单位矩阵.
- 设[tex=0.5x1.214]gNOHIx2AGu3qP//Yn7oxrg==[/tex]是环[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]到环[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]的同态, 证明:[tex=2.929x1.357]7sm0+A17+tx/lVOuO5S85F70wS+QwHOEHbE76/O5U/A=[/tex]是[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]的理想.
- 设[tex=2.786x1.357]FjXX3zhvxUYhb/kCMCOvZw==[/tex] 是一个加群. 定义 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 上的乘法运算为[p=align:center][tex=8.929x1.214]mwVSR6rB8ETCmgrBOZBfKC4aHESn61kUbnYwMS+t5bgAmPHK5UFN6E/t4QuDSXF/[/tex]证明: [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 关于加法和乘法构成一个环.
- 设[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]是一个环，并且[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]对于加法来说作成一个循环群，证明[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]是一个交换环。
- 设 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 是一个交换环. 令 [tex=22.714x1.357]Ox9qLjqRljsr8zy3ePvBrDWTtOx362qRNDpflc+WVBVHtZHYW94HuuaDOWU0hPrFuE/UpoO4Zl/X3hnab7I6R6G7hpXA61s5I/pCzHhVS96KZtqvem5DAON9NJVsKPqDNNIOQC4Vq7VmLOdy4jC3kQ==[/tex]是系数在 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]上的一元多项式的集合. 按通常多项式的加法和乘法定义[tex=1.929x1.357]d5PlggfPq7IWhxnCFu/8ng==[/tex] 中的加法和乘法. 证明 [tex=2.5x1.357]jLtTNY8xgtIxCzUe7Pl3VQ==[/tex]按这样规定的运算构成一个交换环.
内容
- 0
设环 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 对加法作成一个循环群，证明 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 是交换环。
- 1
设 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 是有单位元的环. 证明: 环 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 的可逆元全体 [tex=2.286x1.357]VSrq2EBbjY/lzOCsf2jcIg==[/tex] 关于环 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 的乘法构成群.
- 2
令[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]是由数域[tex=0.786x1.286]BlkXDnmzWHxe4M6E9LlofQ==[/tex]上一切形如[tex=5.0x2.786]jcCMHflCR8OS9TosV6N5vCJqTJglpGfpXaEtTjU9Hwwc+yuMmTz9DJiCpCAUqu2AnZyZpZuHDSyRmftXjpFJnQ==[/tex]的二阶方阵作成的集合.问：[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]对矩阵的普通加法与乘法是否作成环或域?
- 3
设[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]是一个只有有限多个元素的交换环，且[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]没有零因子。证明[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]是一个域。
- 4
设[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]是一个环，在[tex=2.714x1.143]eUSUOmZqe3h2PL1S4fH02FGTr8WV5z6ztlKrlcoXhSM=[/tex]中定义加法及乘法为[tex=11.214x1.357]/teWOzvlACn0WUZjQogaWI+8DYnPCdnCShYQAajeZGA=[/tex]，[tex=13.143x1.357]dgzfoCyYq/iY+ZTOvM+gDdbbQ83gDko/mDo0x3LQ5qFYgDH5Ve6ZL0/wjTtFagNA[/tex]，[tex=3.143x1.214]npydsCcdYUholqtFHoI6bBb+98k8AQvRNvGlE0pP1fM=[/tex]，[tex=2.857x1.214]w/+Nr+NFAwmqP3Kfm8dewQ==[/tex]。证明[tex=2.714x1.143]eUSUOmZqe3h2PL1S4fH02FGTr8WV5z6ztlKrlcoXhSM=[/tex]是一个幺环且有一双边理想与 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex]同构。