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Theorem 2.2.44 Let R be a reduced left morphic ring, if R satisfies ascending chain condition on right annihilators, then every maxmal left ideal of R is a direct summmand of R.
定理2.2.44设R是约化的左morphic环，若R对右零化子满足升链条件，则R的每一个极大左理想L都是R的一个直和项。
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In this paper, we study structure-preserving algorithms for some doubly structured matrices and structured backward errors by Van Loans squared reduced method, quasi-QR factorization and Wielandt-Hoffman theorem.
本文利用Van Loan的平方约化方法，拟QR分解，Wielandt-Hoffman定理等对一些双结构矩阵特征值算法和结构向后误差进行了研究。
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The following statement are the main results:Theorem 2.2.14 Let R be a reduced left morphic ring, then every left ideal of R is generated by a central idempotent. Hence every left ideal of R is also an ideal.
主要结果：定理2.2.14：设R是约化的左morphic环，则R的每一个左理想L都由中心幂等元生成，因而L也是理想。
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Using the dual characteristic pairs, we give the if and only if conditions for which a maximally isotropic subbundle of the double of a Jacobi bialgebroid is a Jacobi-Dirac structure.
在此基础上，建立Jacobi流形的两类对应约化定理，并且给出了约化Jacobi流形上的约化Jacobi结构的具体形式以及其与原流形上的Jacobi结构之间的对应关系。
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We introduce the notion of characteristic pairs and give the definition of dual characteristic pairs of Dirac structures on Lie bialgebroids. Using the dual characteristic pairs, we give the if and only if conditions for which a maximally isotropic subbundle of the double of a Lie bialgebroid is a Dirac structure.
本文在李双代数胚上，引入了Dirac结构的特征对并给出对偶特征对的概念，利用对偶特征对，给出李双代数胚double的极大迷向子丛是Dirac结构的充要条件；其次，分别利用特征对与对偶特征对，将可约Dirac结构分为第一类可约与第二类可约，在此基础上，建立Poisson流形的两类对应约化定理。
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Theorem 2.2.34 Let R be a reduced left morphic ring, then R is a left semi-hereditary ring.
定理2.2.34设R是约化的左morphic环，则R是左半遗传环。
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Theorem 2.2.23 Let R be a reduced left morphic ring, then R is a subdirect product of division rings.
定理2.2.23：设R是约化的左morphic环，则R是除环的亚直积。
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Theorem 2.3.7 Let R be a reduced left morphic ring, then R is a right FS-ring.
定理2.3.7：设R是约化的左morphic环，则R是右FS-环。
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Theorem 2.3.3 Let R be a reduced left morphic ring, then R is a right PS-ring.
定理2.3.3：设R是约化的左morphic环，则R是右PS-环。
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reduction of a fraction：约分
reduction formula 归约公式 | reduction of a fraction 约分 | reduction theorem 化归定理