最后更新于3年前
若<R,+><R,+><R,+>构成交换群，<R,⋅><R,\cdot><R,⋅>构成半群（∀a,b,c∈R,(ab)c=a(bc)\forall a,b,c\in R,\left(ab\right)c=a\left(bc\right)∀a,b,c∈R,(ab)c=a(bc)），⋅\cdot⋅关于+++适合分配律（∀a,b,c∈R,(a+b)c=ac+bc,a(b+c)=ab+ac\forall a,b,c\in R,\left(a+b\right)c=ac+bc,a\left(b+c\right)=ab+ac∀a,b,c∈R,(a+b)c=ac+bc,a(b+c)=ab+ac），则<R,+,⋅><R,+,\cdot><R,+,⋅>为环
交换环：∀a,b∈R,a⋅b=b⋅a\forall a,b\in R,a\cdot b=b\cdot a∀a,b∈R,a⋅b=b⋅a
含幺环：∃e=1R,∀a∈R,a⋅1R=1R⋅a=a\exists e=1_R,\forall a\in R,a\cdot 1_R=1_R\cdot a=a∃e=1R,∀a∈R,a⋅1R=1R⋅a=a
非零元aaa为左零因子：∃b∈R,b≠0,ab=0\exists b\in R,b\neq 0,ab=0∃b∈R,b=0,ab=0
零因子：aaa同时为左零因子和右零因子，RRR为零因子环
aaa为左逆元：∃b∈R,ab=1R\exists b\in R,ab=1_R∃b∈R,ab=1R
逆元：aaa同时为左逆元和右逆元
整环：RRR为交换环、含幺环、无零因子环
∀a∈R,0a=a0=0\forall a\in R,0a=a0=0∀a∈R,0a=a0=0
∀a,b∈R,(−a)b=a(−b)=−ab\forall a,b\in R,\left(-a\right)b=a\left(-b\right)=-ab∀a,b∈R,(−a)b=a(−b)=−ab
∀a,b∈R,(−a)(−b)=ab\forall a,b\in R,\left(-a\right)\left(-b\right)=ab∀a,b∈R,(−a)(−b)=ab
∀n∈Z,∀a,b∈R,(nab)=a(nb)=nab\forall n\in Z,\forall a,b\in R,\left(nab\right)=a\left(nb\right)=nab∀n∈Z,∀a,b∈R,(nab)=a(nb)=nab
\forall a_i,b_j\in R,\left(\sum_\limits{i=1}^{n}{a_i}\right)\left(\sum_\limits{j=1}^{n}{b_j}\right)=\sum_\limits{i=1}^{n}{\sum_\limits{j=1}^{n}{a_ib_j}}