What will be the smallest ring containing two rings?












5















Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?



In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?




I need some explanations to this. Thank you.






abstract-algebra ring-theory commutative-algebra integral-domain







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    5















    Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?



    In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?




    I need some explanations to this. Thank you.






    abstract-algebra ring-theory commutative-algebra integral-domain







    share|cite|improve this question



























      5












      5








      5


      1






      Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?



      In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?




      I need some explanations to this. Thank you.






      abstract-algebra ring-theory commutative-algebra integral-domain







      share|cite|improve this question
















      Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?



      In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?




      I need some explanations to this. Thank you.






      abstract-algebra ring-theory commutative-algebra integral-domain




      abstract-algebra ring-theory commutative-algebra integral-domain






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 7 hours ago









      Zvi

      4,855430




      4,855430










      asked 11 hours ago









      user371231

      692511




      692511






















          1 Answer
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          6














          The smallest ring containing both $R$ and $S$ is the set
          $$ left{sum_{i=1}^n r_is_i Bigg| nin mathbb{N}, r_iin R, s_i in Sright}. $$



          Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbb{Z}[x,y]/(xy)$, $R= mathbb{Z}[x]$ and $S = mathbb{Z}[y]$.






          share|cite|improve this answer























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            1 Answer
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            active

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            6














            The smallest ring containing both $R$ and $S$ is the set
            $$ left{sum_{i=1}^n r_is_i Bigg| nin mathbb{N}, r_iin R, s_i in Sright}. $$



            Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbb{Z}[x,y]/(xy)$, $R= mathbb{Z}[x]$ and $S = mathbb{Z}[y]$.






            share|cite|improve this answer




























              6














              The smallest ring containing both $R$ and $S$ is the set
              $$ left{sum_{i=1}^n r_is_i Bigg| nin mathbb{N}, r_iin R, s_i in Sright}. $$



              Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbb{Z}[x,y]/(xy)$, $R= mathbb{Z}[x]$ and $S = mathbb{Z}[y]$.






              share|cite|improve this answer


























                6












                6








                6






                The smallest ring containing both $R$ and $S$ is the set
                $$ left{sum_{i=1}^n r_is_i Bigg| nin mathbb{N}, r_iin R, s_i in Sright}. $$



                Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbb{Z}[x,y]/(xy)$, $R= mathbb{Z}[x]$ and $S = mathbb{Z}[y]$.






                share|cite|improve this answer














                The smallest ring containing both $R$ and $S$ is the set
                $$ left{sum_{i=1}^n r_is_i Bigg| nin mathbb{N}, r_iin R, s_i in Sright}. $$



                Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=mathbb{Z}[x,y]/(xy)$, $R= mathbb{Z}[x]$ and $S = mathbb{Z}[y]$.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 7 hours ago









                Zvi

                4,855430




                4,855430










                answered 11 hours ago









                Pierre-Guy Plamondon

                8,65011639




                8,65011639






























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