  
  [1X3 [33X[0;0YIdeals and left ideals[133X[101X
  
  [33X[0;0YIn  this  section  we  describe several functions related to ideals and left
  ideals of skew braces. References: [GV17] and [SV18].[133X
  
  
  [1X3.1 [33X[0;0YLeft ideals[133X[101X
  
  [33X[0;0YAn left ideal [23XI[123X of a skew brace [23XA[123X is a subgroup [23XI[123X of the additive group of [23XA[123X
  such that [23X\lambda_a(I)\subseteq I[123X for all [23Xa\in A[123X.[133X
  
  [1X3.1-1 LeftIdeals[101X
  
  [33X[1;0Y[29X[2XLeftIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list with the left ideals of the skew brace [3Xobj[103X[133X
  
  [1X3.1-2 StrongLeftIdeals[101X
  
  [33X[1;0Y[29X[2XStrongLeftIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya  list with the left ideals of the skew brace [3Xobj[103X that are normal
            in the additive group of [23XA[123X[133X
  
  [1X3.1-3 IsLeftIdeal[101X
  
  [33X[1;0Y[29X[2XIsLeftIdeal[102X( [3Xobj[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the subset is a left ideal of [3Xobj[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(8,4);[127X[104X
    [4X[28X<brace of size 8>[128X[104X
    [4X[25Xgap>[125X [27Xleftideals := LeftIdeals(br);[127X[104X
    [4X[28X[ <brace of size 1>, <brace of size 2>, <brace of size 4>, <brace of size 8> ][128X[104X
    [4X[25Xgap>[125X [27XList(leftideals, x->IsLeftIdeal(br, x));[127X[104X
    [4X[28X[ true, true, true, true ][128X[104X
    [4X[25Xgap>[125X [27XList(leftideals, IdBrace);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 2, 1 ], [ 4, 1 ], [ 8, 4 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YIdeals[133X[101X
  
  [33X[0;0YAn ideal [23XI[123X of a skew brace [23XA[123X is a normal subgroup [23XI[123X of the additive group of
  [23XA[123X such that [23X\lambda_a(I)\subseteq I[123X and [23Xa\circ I=I\circ a[123X for all [23Xa\in A[123X.[133X
  
  [1X3.2-1 IsIdeal[101X
  
  [33X[1;0Y[29X[2XIsIdeal[102X( [3Xobj[103X, [3Xsubset[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the [3Xsubset[103X is a left ideal of [3Xobj[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(8,4);[127X[104X
    [4X[28X<brace of size 8>[128X[104X
    [4X[25Xgap>[125X [27Xleftideals := LeftIdeals(br);[127X[104X
    [4X[28X[ <brace of size 1>, <brace of size 2>, <brace of size 4>, <brace of size 8> ][128X[104X
    [4X[25Xgap>[125X [27XList(leftideals, x->IsLeftIdeal(br, x));[127X[104X
    [4X[28X[ true, true, true, true ][128X[104X
    [4X[25Xgap>[125X [27XList(leftideals, IdBrace);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 2, 1 ], [ 4, 1 ], [ 8, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.2-2 Ideals[101X
  
  [33X[1;0Y[29X[2XIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list with the ideals of the skew brace [3Xobj[103X[133X
  
  [1X3.2-3 AsIdeal[101X
  
  [33X[1;0Y[29X[2XAsIdeal[102X( [3Xarg1[103X, [3Xarg2[103X ) [32X operation[133X
  
  [1X3.2-4 IdealGeneratedBy[101X
  
  [33X[1;0Y[29X[2XIdealGeneratedBy[102X( [3Xobj[103X, [3Xsubset[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe ideal of [3Xobj[103X generated by the given [3Xsubset[103X[133X
  
  [33X[0;0YThe  ideal  of a skew brace [23XA[123X generated by a subset [23XX[123X is the intersection of
  all the ideals of [23XA[123X containing [23XX[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(6,6);;[127X[104X
    [4X[25Xgap>[125X [27XAsList(br);[127X[104X
    [4X[28X[ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)>, <(1,4)(2,5)(3,6)>, [128X[104X
    [4X[28X  <(1,5,3,4,2,6)>, <(1,6,2,4,3,5)> ][128X[104X
    [4X[25Xgap>[125X [27XIdealGeneratedBy(br, [last[2]]);[127X[104X
    [4X[28X<brace of size 3>[128X[104X
  [4X[32X[104X
  
  [1X3.2-5 IntersectionOfTwoIdeals[101X
  
  [33X[1;0Y[29X[2XIntersectionOfTwoIdeals[102X( [3Xideal1[103X, [3Xideal2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe intersection of [3Xideal1[103X and [3Xideal2[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(6,6);;[127X[104X
    [4X[25Xgap>[125X [27XIdeals(br);[127X[104X
    [4X[28X[ <brace of size 6>, <brace of size 2>, <brace of size 3>, <brace of size 1> ][128X[104X
    [4X[25Xgap>[125X [27XIntersectionOfTwoIdeals(last[2],last[3]);[127X[104X
    [4X[28X<brace of size 1>[128X[104X
  [4X[32X[104X
  
  [1X3.2-6 SumOfTwoIdeals[101X
  
  [33X[1;0Y[29X[2XSumOfTwoIdeals[102X( [3Xideal1[103X, [3Xideal2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe sum of [3Xideal1[103X and [3Xideal2[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(6,6);;[127X[104X
    [4X[25Xgap>[125X [27XIdeals(br);[127X[104X
    [4X[28X[ <brace of size 6>, <brace of size 2>, <brace of size 3>, <brace of size 1> ][128X[104X
    [4X[25Xgap>[125X [27XSumOfTwoIdeals(last[2],last[3]);[127X[104X
    [4X[28X<brace of size 6>[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YSequences (left) ideals[133X[101X
  
  [1X3.3-1 LeftSeries[101X
  
  [33X[1;0Y[29X[2XLeftSeries[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe left ideals of the left series of [3Xobj[103X[133X
  
  [33X[0;0YThe  left  series  of  a  skew  brace  [23XA[123X is defined recursively as [23XA^1=A[123X and
  [23XA^{n+1}=A*A^n[123X  for  [23Xn\geq1[123X,  where  [23Xa*b=\lambda_a(b)-b[123X.  Each  [23XA^n[123X is a left
  ideal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(8,20);[127X[104X
    [4X[28X<skew brace of size 8>[128X[104X
    [4X[25Xgap>[125X [27XLeftSeries(br);[127X[104X
    [4X[28X[ <skew brace of size 8>, <brace of size 2>, <brace of size 1> ][128X[104X
  [4X[32X[104X
  
  [1X3.3-2 RightSeries[101X
  
  [33X[1;0Y[29X[2XRightSeries[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe ideals of the right series of [3Xobj[103X[133X
  
  [33X[0;0YThe  right series of a skew brace 0[23XA[123X is defined recursively as [23XA^{(1)}=A[123X and
  [23XA^{(n+1)}=A*A^{(n)}[123X for [23Xn\geq1[123X, where [23Xa*b=\lambda_a(b)-b[123X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(8,20);[127X[104X
    [4X[28X<skew brace of size 8>[128X[104X
    [4X[25Xgap>[125X [27XRightSeries(br);[127X[104X
    [4X[28X[ <skew brace of size 8>, <brace of size 2>, <brace of size 1> ][128X[104X
  [4X[32X[104X
  
  [1X3.3-3 IsLeftNilpotent[101X
  
  [33X[1;0Y[29X[2XIsLeftNilpotent[102X( [3Xobj[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the skew brace [3Xobj[103X is left nilpotent.[133X
  
  [33X[0;0YA skew brace [23XA[123X is said to be left nilpotent if there exists [23Xn\geq1[123X such that
  [23XA^n=0[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsLeftNilpotent(SmallBrace(8,18));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLeftNilpotent(SmallBrace(12,2));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X3.3-4 IsSimpleSkewbrace[101X
  
  [33X[1;0Y[29X[2XIsSimpleSkewbrace[102X( [3Xobj[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the skew brace [3Xobj[103X is simple.[133X
  
  [33X[0;0YA skew brace [23XA[123X is said to be simple if [23X\{0\}[123X and [23XA[123X are its only ideals.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsSimple(SmallSkewbrace(12,22));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSimple(SmallSkewbrace(12,21));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X3.3-5 IsRightNilpotent[101X
  
  [33X[1;0Y[29X[2XIsRightNilpotent[102X( [3Xobj[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the skew brace [3Xobj[103X is right nilpotent.[133X
  
  [33X[0;0YA  skew  brace  [23XA[123X  is said to be right nilpotent if there exists [23Xn\geq1[123X such
  that [23XA^{(n)}=0[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsRightNilpotent(SmallBrace(8,18));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsRightNilpotent(SmallBrace(12,2));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.3-6 LeftNilpotentIdeals[101X
  
  [33X[1;0Y[29X[2XLeftNilpotentIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe list of right or left nilpotent ideals of [3Xobj[103X[133X
  
  [33X[0;0YAn ideal [23XI[123X of a skew brace [23XA[123X is said to be left if it is left nilpotent as a
  skew brace.[133X
  
  [1X3.3-7 RightNilpotentIdeals[101X
  
  [33X[1;0Y[29X[2XRightNilpotentIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe list of right or left nilpotent ideals of [3Xobj[103X[133X
  
  [33X[0;0YAn  ideal [23XI[123X of a skew brace [23XA[123X is said to be right nilpotent if An ideal [23XI[123X of
  a skew brace [23XA[123X is said to be left if it is right nilpotent as a skew brace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(8,18);;[127X[104X
    [4X[25Xgap>[125X [27XIsLeftNilpotent(br);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLeftNilpotentIdeals(br);[127X[104X
    [4X[28X[ <brace of size 8>, <brace of size 4>, <brace of size 1> ][128X[104X
    [4X[25Xgap>[125X [27XIsRightNilpotent(br);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XRightNilpotentIdeals(br);[127X[104X
    [4X[28X[ <brace of size 4>, <brace of size 1> ][128X[104X
  [4X[32X[104X
  
  [1X3.3-8 SmoktunowiczSeries[101X
  
  [33X[1;0Y[29X[2XSmoktunowiczSeries[102X( [3Xobj[103X, [3Xbound[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list of [3Xbound[103X left ideals of the Smoktunowicz's series of [3Xobj[103X[133X
  
  [33X[0;0YThe  Smoktunowicz's  series  of  a  skew  brace  [23XA[123X is defined recursively as
  [23XA^{[1]}=A[123X  and  [23XA^{[n+1]}[123X  is  the  additive  subgroup  of  [23XA[123X  generated  by
  [23XA^{[i]}*A^{[n+1-i]}[123X for [23X1\leq i+j\leq n+1[123X, where [23Xa*b=\lambda_a(b)-b[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(16,145);;[127X[104X
    [4X[25Xgap>[125X [27XSmoktunowiczSeries(br,4);[127X[104X
    [4X[28X[ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,[128X[104X
    [4X[28X  <brace of size 2> ][128X[104X
    [4X[25Xgap>[125X [27XSmoktunowiczSeries(br,5);[127X[104X
    [4X[28X[ <brace of size 16>, <brace of size 8>, <brace of size 4>, <brace of size 2>,[128X[104X
    [4X[28X  <brace of size 2>, <brace of size 1> ][128X[104X
  [4X[32X[104X
  
  [1X3.3-9 Socle[101X
  
  [33X[1;0Y[29X[2XSocle[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe ideals of the socle series of [3Xobj[103X[133X
  
  [33X[0;0YThe socle of a skew brace [23XA[123X is the ideal [23X\ker\lambda\cap Z(A,+)[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSocle(SmallSkewbrace(6,2));[127X[104X
    [4X[28X<brace of size 1>[128X[104X
    [4X[25Xgap>[125X [27XSocle(SmallBrace(8,20));[127X[104X
    [4X[28X<brace of size 8>[128X[104X
    [4X[25Xgap>[125X [27XSocle(SmallBrace(8,2));[127X[104X
    [4X[28X<brace of size 4>[128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YMutipermutation skew braces[133X[101X
  
  [1X3.4-1 SocleSeries[101X
  
  [33X[1;0Y[29X[2XSocleSeries[102X( [3Xobj[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe socle series of [3Xobj[103X[133X
  
  [33X[0;0YThe  socle  series  of  a  skew  brace [23XA[123X is defined recursively as [23XA_1=A[123X and
  [23XA_{n+1}=A_n/\mathrm{Soc}(A_n)[123X, see [SV18].[133X
  
  [1X3.4-2 MultipermutationLevel[101X
  
  [33X[1;0Y[29X[2XMultipermutationLevel[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe multipermutation level of the skew brace [3Xobj[103X[133X
  
  [33X[0;0YThe  multipermutation  level  of  a  skew brace [23XA[123X is defined as the smallest
  positive  integer [23Xn[123X such that the [23Xn[123X-th term [23XA_n[123X of the socle series has only
  one element, see Definition 5.17 of [SV18].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(8,20);;[127X[104X
    [4X[25Xgap>[125X [27XSocleSeries(br);[127X[104X
    [4X[28X[ <brace of size 8>, <brace of size 1> ][128X[104X
    [4X[25Xgap>[125X [27XMultipermutationLevel(br);[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X3.4-3 IsMultipermutation[101X
  
  [33X[1;0Y[29X[2XIsMultipermutation[102X( [3Xobj[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X  if  the skew brace [3Xobj[103X has finite multipermutation level and
            [3Xfalse[103X otherwise[133X
  
  [1X3.4-4 Fix[101X
  
  [33X[1;0Y[29X[2XFix[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X  if  the skew brace [3Xobj[103X has finite multipermutation level and
            [3Xfalse[103X otherwise[133X
  
  [1X3.4-5 KernelOfLambda[101X
  
  [33X[1;0Y[29X[2XKernelOfLambda[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe  kernel of the map [23X\lambda[123X as a subset of elements of the skew
            brace [3Xobj[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(6,1);;[127X[104X
    [4X[25Xgap>[125X [27XKernelOfLambda(br);[127X[104X
    [4X[28X[ <()>, <(1,2,3)(4,5,6)>, <(1,3,2)(4,6,5)> ][128X[104X
  [4X[32X[104X
  
  [1X3.4-6 Quotient[101X
  
  [33X[1;0Y[29X[2XQuotient[102X( [3Xobj[103X, [3Xideal[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe quotient [3Xobj[103X by [3Xideal[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(8,10);;[127X[104X
    [4X[25Xgap>[125X [27Xideals := Ideals(br);;[127X[104X
    [4X[25Xgap>[125X [27XQuotient(br, ideals[5]);[127X[104X
    [4X[28X<brace of size 4>[128X[104X
    [4X[25Xgap>[125X [27Xbr/ideals[2];[127X[104X
    [4X[28X<brace of size 2>[128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YPrime and semiprime ideals[133X[101X
  
  [1X3.5-1 IsPrimeBrace[101X
  
  [33X[1;0Y[29X[2XIsPrimeBrace[102X( [3Xobj[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the skew brace [3Xobj[103X is prime[133X
  
  [33X[0;0YA  skew  brace  [23XA[123X is said to be prime if for all non-zero ideals [23XI[123X and [23XJ[123X one
  has [23XI*J\ne 0[123X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimeBrace(SmallBrace(24,12));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPrimeBrace(SmallBrace(24,94));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.5-2 IsPrimeIdeal[101X
  
  [33X[1;0Y[29X[2XIsPrimeIdeal[102X( [3Xobj[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the ideal [3Xobj[103X is prime[133X
  
  [33X[0;0YAn  ideal  [23XI[123X  of  a  skew brace [23XA[123X is said to be prime if [23XA/I[123X is a prime skew
  brace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallBrace(24,94);[127X[104X
    [4X[28X<brace of size 24>[128X[104X
    [4X[25Xgap>[125X [27XIsPrimeBrace(br);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdeals(br);[127X[104X
    [4X[28X[ <brace of size 24>, <brace of size 1> ][128X[104X
    [4X[25Xgap>[125X [27XIsPrimeIdeal(last[2]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.5-3 PrimeIdeals[101X
  
  [33X[1;0Y[29X[2XPrimeIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe list of prime ideals of the skew brace [3Xobj[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrimeIdeals(SmallBrace(24,94));[127X[104X
    [4X[28X[ <brace of size 24>, <brace of size 1> ][128X[104X
  [4X[32X[104X
  
  [1X3.5-4 IsSemiprime[101X
  
  [33X[1;0Y[29X[2XIsSemiprime[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the skew brace [3Xobj[103X is semiprime[133X
  
  [33X[0;0YAn  ideal  [23XI[123X of a skew brace [23XA[123X is said to be semiprime if [23XA/I[123X is a semiprime
  skew brace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := DirectProductSkewbraces(SmallSkewbrace(12,22),SmallSkewbrace(12,22));;[127X[104X
    [4X[25Xgap>[125X [27XIsSemiprime(br);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.5-5 IsSemiprimeIdeal[101X
  
  [33X[1;0Y[29X[2XIsSemiprimeIdeal[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the ideal [3Xobj[103X is semiprime[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSemiprimeIdeals(SmallSkewbrace(12,24));[127X[104X
    [4X[28X[ <skew brace of size 12> ][128X[104X
    [4X[25Xgap>[125X [27XIsSemiprimeIdeal(last[1]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.5-6 SemiprimeIdeals[101X
  
  [33X[1;0Y[29X[2XSemiprimeIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe list of semiprime ideals of the skew brace [3Xobj[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSemiprimeIdeals(SmallSkewbrace(12,22));[127X[104X
    [4X[28X[ <skew brace of size 12>, <brace of size 1> ][128X[104X
    [4X[25Xgap>[125X [27XSemiprimeIdeals(SmallSkewbrace(12,24));[127X[104X
    [4X[28X[ <skew brace of size 12> ][128X[104X
  [4X[32X[104X
  
  [1X3.5-7 BaerRadical[101X
  
  [33X[1;0Y[29X[2XBaerRadical[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe Baer radical of the skew brace [3Xobj[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(6,2);;[127X[104X
    [4X[25Xgap>[125X [27XBaerRadical(br);[127X[104X
    [4X[28X<skew brace of size 6>[128X[104X
  [4X[32X[104X
  
  [1X3.5-8 IsBaer[101X
  
  [33X[1;0Y[29X[2XIsBaer[102X( [3Xobj[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the skew brace [3Xobj[103X is ia Baer radical skew brace.[133X
  
  [33X[0;0YA  skew brace [23XA[123X is said to be Baer radical if [23XA=B(A)[123X, where [23XB(A)[123X is the Baer
  radical of [23XA[123X (i.e., the intersection of all prime ideals of [23XA[123X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(6,2);;[127X[104X
    [4X[25Xgap>[125X [27XIsBaer(br);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.5-9 WedderburnRadical[101X
  
  [33X[1;0Y[29X[2XWedderburnRadical[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe Wedderburn radical of the skew brace [3Xobj[103X[133X
  
  [33X[0;0YThe  Wedderburn radical of a skew brace is the intersection of all its prime
  ideals[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(6,2);;[127X[104X
    [4X[25Xgap>[125X [27XWedderburnRadical(br);[127X[104X
    [4X[28X<brace of size 3>[128X[104X
  [4X[32X[104X
  
  [1X3.5-10 SolvableSeries[101X
  
  [33X[1;0Y[29X[2XSolvableSeries[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list with the solvable series of the skew brace [3Xobj[103X[133X
  
  [33X[0;0YThe  solvable series of a skew brace [23XA[123X is defined recursively as [23XA_{1}=A[123X and
  [23XA_{n+1}=A_{n}*A_{n}[123X for [23Xn\geq1[123X, where [23Xa*b=\lambda_a(b)-b[123X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(8,20);;[127X[104X
    [4X[25Xgap>[125X [27XIsSolvable(br);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSolvableSeries(br);[127X[104X
    [4X[28X[ <skew brace of size 8>, <brace of size 2>, <brace of size 1> ][128X[104X
    [4X[25Xgap>[125X [27Xbr := SmallSkewbrace(12,23);;[127X[104X
    [4X[25Xgap>[125X [27XIsSolvable(br);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X3.5-11 IsMinimalIdeal[101X
  
  [33X[1;0Y[29X[2XIsMinimalIdeal[102X( [3Xobj[103X, [3Xideal[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if [3Xideal[103X is a minimal ideal of [3Xobj[103X An ideal [23XI[123X of [23XA[123X is said to
            be  [13Xminimal[113X  if does not contain any other ideal of [23XA[123X. To check if
            an  ideal [23XI[123X of [23XA[123X is minimal, one computes the ideals of [23XI[123X and keep
            only those that are simple as a skew brace.[133X
  
  [1X3.5-12 MinimalIdeals[101X
  
  [33X[1;0Y[29X[2XMinimalIdeals[102X( [3Xobj[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list of minimal ideals of the skew brace [3Xobj[103X[133X
  
